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PRACTICAL TREATISE 



GEARING 



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II 




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PROVIDENCE, R. [. 

BROWN & SHARPE MANUFACTURING COMPANY. 

188G. 



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Copyright 1886, by 
BBOWN & SHARPE MFG. CO. 



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tmp96 026436 



Press of J. W. Pratt & Son, 73 to 79 Fulton Street, New York. 



PREFACE. 

This Book is made for men in practical life ; for those who 
would like to know how to construct gear wheels, but whose 
duties do not afford them sufficient leisure for acquu'ing a 
technical knowledge of the subject. 



CONTENTS 

PART I. 

Chapter I. 



PAGE. 



Pitch Cii-cle — Pitch — Tooth — Space — Addendum or Face- 
Flank — Clearance 1 

Chapter II. 
Classification — Sizing Blanks and Tooth Parts from Circular 

Pitch — Center Distance 5 

Chapter III. 
Single Curve Gears of 30 Teeth and over 9 

Chapter IV. 

Rack to Mesh with Single Curve Gears having 30 Teeth and 

over 12 

Chapter V. 
Diametral Pitch — Sizing Blanks and Teeth — Distance between 

Centers of Wheels 16 

Chapter YI. 
Single-Curve Gears, having Less than 30 Teeth — Gears and 

Racks to Mesh with Gears having Less than 30 Teeth. . . 20 

Chapter VII. 
Double-Curve Teeth— Gear of 15 Teeth— Rack 25 

Chapter VIII. 
Double-Curve Gears, having More and Less than 15 Teeth — 

Annular Gears 30 

Chapter IX. 
Bevel Gear Blanks 34 

Chapter X. 
Bevel Gears—Form and Size of Teeth— Cutting Teeth 38 



VI CONTENTS. 

Chapter XI. 

PAGE. 

Worm Wheels — Sizing Blanks of 32 Teeth and over. 42 

Chapter XII. 

Sizing Gears when the Distance between Centers and the 
Katio of Speeds are fixed — General Remarks — Width of 
Face of Spur Gears — Sj^eed of Gear Cutters — Table of 
Tooth Parts 57 



PART II. 



Chapter I. 
Tangent of Arc and Angle 69 

Chapter II. 
Application of Table of Tangents to Problems occurring in 

Machine Construction — Angles of Bevel Gears 72 

Chapter III. 
Sine, Cosine and Secant — Some of their Applications in 

Machine Construction 77 

Chapter IV. 
Further Application of Circular Functions — Whole Diameter 

of Bevel Gear Blanks — Face Angle of Bevel Gear Blanks 82 

Chapter V. 
Spkal Gears — Calculations for Pitch of Spirals 87 

Chapter VI. 
Examples in Calculations of Pitch of Spu^als — Angle of 
Spiral — Cii'cumference of Spiral Gears — A few Hints on 
Cutting 91 

Chapter VII. 
Normal Pitch of Spiral Gears — Curvature of Pitch Surface — 

Formation of Cutters 96 



CONTENTS. Vll 

Chapter VIII. 

PAGE. 

Screw Gears and Spiral Gears — General Remarks 102. 

Chapter IX. 

Continued Fractions — Some Applications in Machine Con- 
struction 104 

Chapter X. 
Angle of Pressure 109 

Chapter XI. 
Internal Gears — Tables — Index Ill 



INDEX. 



A. 

PAGE. 

Abbreviations of Parts of Teeth and Gears 4 

Addendum 2 

Angle, How to Lay Off an 74, 85 

" Increment 84 

" of Edge 72 

" of Face 72 

'^ of Pressure 109 

*' of Spiral 90 

Angular Velocity 1 

Annular Gears 32, 111 

Arc of Action 110 

B. 

Base Circle 11 

" of Epicycloidal System 25 

" of Internal Gears Ill 

Bevel Gear Blanks , 34 

" Angles by Diagram 36 

" " by Calculation 72, 84 

Form of Teeth of 38 

" Whole Diameter of 37, 82 

C. 

Centers, Line of 2 

Circular Pitch 4 

Classification of Gearing 5 

Clearance at Bottom of Space 6 

" in Pattern Gears 8 

Condition of Constant Velocity Ratio 2 

Contact, Arc of 110 

Continued Fractions 104 

Coppering Solution 63 



INDEX. IX 

PAGE. 

Cutters, How to Order 61 

" Table of Epicycloidal 02 

" " of Involute GO 

" of Speeds for 59 

D. 

Decimal Equivalents, Tables of 114 

Diameter Increment 82 

of Pitch Circle G 

Pitcli 5 

Diametral Pitcli 17 

Distance between Centers 8 

E. 

Elements of Gear Teetli = . . . . 5 

Epicycloidal Gears, with more and less than 15 Teeth 30 

" with 15 Teeth 25 

Rack 27 

F. 

Face, Width of Spur Gear 58 

Flanks of Teeth in Low-numbered Pinions 20 

G. 

Gear Cutters, How to Order 6] 

" Patterns 8 

Gearing Classified 5 

Gears, Bevel 34, 73, 82 

" Epicycloidal 25 

'• Involute 9 

" Sph-al 87 

" Worm 42 

H. 

Herring-bone Gears 103 

I. 

Increment, Angle , 84 

Diameter 82 

Interchangeable Gears 24 

Internal or Annular Gears Ill 

Involute Gears, 30 Teeth and over 9 

'' " with Less than 30 Teeth 20 

Rack 12 



X INDEX. 

L. 

PAGE, 

Limiting Numbers of Teeth in Internal Gears 32 

Line of Centers 2 

" of Pressure , 12, 109 

Linear Velocity 1 

N. 

Normal 96 

" Helix 96 

" Pitch 96 

O. 

Odontograph, Willis 113 

Original Cylinders 1 

P. 

Pattern Gears 8 

Pitch Circle 3 

" Circular or Linear 4 

" a Diameter 6 

" Diametral 17 

" Normal 96 

" of Spirals 90 

Polygons, Calculations for Diameters of 79 

E. 

Kack 12 

" for Epicycloidal Gears 27 

*' for Livolute " 12 

" for Spiral " 101 

Relative Angular Velocity 2 

Rolling Contact of Pitch Circle. 3 

S. 

Screw Gearing 87, 103 

Single-Curve Teeth 9 

Speed of Gear Cutters 59 

Spiral Gearing 87 

Standard Templets 27 



INDEX. XI 

T. 

PAGE. 

Table of Decimal Equivalents 114 

" of Sines, etc IIG 

" of Speeds for Gear Cutters 59 

" of Tooth Parts 04 

V. 

Velocity, Angular 2 

" Linear 1 

Kelative 2 

W. 

Wear of Teeth 58, 103 

Wilhs Odontograph 113 

"Worm Gears 42 



PART I. 



CHAPTER I. 

PITCH CIRCLE, PITCH, TOOTH, SPACE, ADDENDUM OR FACE, FLANK, 

CLEARANCE. 



Let two cylinders, Fig. 1, toucli each other, theh'.^^J^^^^i ^^^' 
axes be parallel and the cylinders be on shafts, turning 
freely. If, now, we turn one cylinder, the adhesion of 
its surface to the surface of the other cylinder will 
make that turn also. The surfaces touching each 
other, without slipping one upon the other, will evi- 
dently move through the same distance in a given 
time. This surface speed is called linear velocity. ty. 

TANGENT CYLINDERS. 




Fi-. 1 



Linear Velocity is the number of inches or feet a 
point moves along a line in a given time. A moving 
belt, a piston of an engine have linear velocity. These 



BEOWN & SHARPE MFG. CO. 



Angular 
locity. 



cylinders in tui'ning about their axes also pass through 
angles, whose vertices are at the axes of the cylin- 
ders. The number of angles passed through in a 
given time is called angular velocity. 

Angular Yelocity is usually expressed by turns in a 

minute. Four right angles make one turn. If these 

two cylinders are of the same size, one wdll make the 

same number of turns in a minute that the other 

Relative An- makes. If oue cylinder is twice as large as the other, 

gular Velocity. *' o ^ 

the smaller will make two turns while the larger makes 
one, but the linear velocity of the surface of each cylin- 
der remains the same. 

This combination would be very useful in mechan- 
ism if we could be sure that one cylinder would always 
turn the other without slipping. 




:Piir. 2 



50.tND^^^ 




Fio;. 3 



Land. 

Addendum. 

Tooth. 

Gear. 

Train. 



In the periphery of these two cylinders, as in Fig. 
2, cut equidistant grooves. In any grooved piece the 
places between grooves are called lands. Upon the 
lands add parts ; these parts are called addenda. A 
land and its addendum is called a tooth. A toothed 
cylinder is called a gear. Two or more gears with 
teeth interlocking are called a train. A line, c c', Fig. 



PROVIDENCE, R. I. 



Addendum 
Circle. 



2 or 3, between the centers of two wheels is called the Line of cen 
lifie of ce?Uers. A circle just touching the addenda 
is called the addendum circle. 

The circumference of the cylinders without teeth is 
called the pitch circle. This circle exists geometri- ^^'^^ circle, 
cally in every gear and is still called the pitch circle p^^.^^^ circle 
or the primitive circle. In the study of gear wheels, it ihe^^^PrimUive 
is the problem to so shape the teeth that the pitch ^^^^^®- 
cii-cles will just touch each other without slipioing. 

On two fixed centers there can turn only two circles, 
one circle on each center, in a given relative angular 
velocity and touch each other without slipping. 




! THICKNESS OF 
WIDTH OF SPACE I TOOTH 

AT PITCH LINE | AT PITCH LINE 




Fig.4r 



4 BROWN & SHARPE MFG. CO. 

Space. The groove between two teeth is called a space. 

In cut gears the width of space at pitch line and thick- 
ness of tooth at pitch line are equal. The distance 
between the center of one tooth and the center of the 

Circular Pitch, next tooth, measured along the pitch line, is the cir- 
cular pitch ; that is, the circular pitch is equal to a 
Tooth Thick- tooth and a space ; hence, the thickness of a tooth at 
the pitch line is equal to one-half the circular pitch. 

ti^s ^of^Parts "^^^ D = diameter of addendum cu'cle. 

for Teeth and a T\' — u u i->ifoli " 

Gear. ^ — ^ViQ^\l 

" P' = circular pitch. 

" ^=: thickness of tooth at pitch line. 

" 5 = addendum or face, also length of working 

part of tooth below pitch line or flank. 
" 2s =D" or twice the addendum, equals the 

working depth of teeth of two gears in 

mesh. 
" y*= clearance or extra depth of space below 

working depth. 
" s -f-y= depth of space below i^itch line. 
" D"+/'=r whole depth of space. 
" N= number of teeth in one gear. 
" ;r— 3.1416 or the cu'cumference when diameter 

is 1. 
P' is read "P prime.'' D" is read "D second." 
It is read, "pi." 

If we multiply the diameter of any circle by ;r, the 
To find the product will be the circumference of this circle. If 
and Diameter we divide the Circumference of any cu'cle by n^ the 
quotient will be the diameter of this circle. 



rEOVlDENCE, R. I. 



CHAPTER 11. 

CLASSIFICATION- SIZING BLANKS AND TOOTH PARTS FROM 
CIRCULAR PITCH— CENTER DISTANCE— PATTERN GEARS. 



If we conceive the pitch of a pair of gears to l^e^be^^^eth® ^^ 
made the smallest possible, we ultimately come to the 
conception of teeth that are merely lines upon the 
original pitch surfaces. These lines are called ele- 
ments of the teeth. Gears may be classified with 
reference to the elements of their teeth, and also with 
reference to the relative position of their axes or -shafts. 
In most gears the elements of teeth are either straight 
lines or helices (screw-like lines). 

In Part I. of this work, we shall treat upon three 

KINDS OF GEARS : 

jPaVs^— Spur Gears ; those connecting parallel shafts ^p"^' oears. 
and whose tooth elements are straight. 

Second — Bevel Gears; those connecting shafts ^e^ei Gears, 
whose axes meet when sufficiently prolonged, and the 
elements of whose teeth are straight lines. In bevel 
gears the surfaces that touch each other, without 
slipping, are upon cones or parts of cones whose 
apexes are at the same point where axes of shafts meet. 

Third — Screw or Worm Gears; those connecting ^^.^^^j^" *^J^^^^° ^ 
shafts that are neither parallel nor meet, and the ele- 
ments of whose teeth are helical or screw-like. 

The circular pitch and number of teeth in a wheel g. . ^ 
being given, the diameter of the wheel and size of i>iank.«, &c. 
tooth parts are found as follows : 

Dividing by 3.1416 is the same as multiplying by 
g7^— . Now 37^41-^= 3183; hence, multiply the cir- 
cumference of a circle by .3183 and the product will be 
the diameter of the circle. Multiply the circular pitch 
by .3183 and the product will be the same part of the 



eter. 



6 BROWN & SHARPE MFG. CO. 

diameter of pitcli cii'cle that the circular pitch is of the 

circumference of pitch circle. This part or modulus 
A Diameter fg called a diameter pit ch. There are as many diameter 

pitches contained in the diameter of j^itch cu'cle as 

there are teeth in the wheel. 
A Diameter Most mechauics make the addendum of teeth equal 

Pitch and the i 

Addendum^Q one diameter pitch. Hence we can designate this 

measurure the ^ o 

same, radiaUy. modulus or diameter pitch by the same letter as we do 
the addendum ; that is, let 5= a diameter pitch. 

.3183 P' — 5, or cu'cular pitch multiplied by .3183=^, 
or a diameter pitch. 

Ns=r>', or number of teeth in a wheel, multiplied 
Pi? h'^'ircie ^^ ^^ ^ diameter pitch, equals diameter of j)itch cii'cle. 

(N + 2) s=D, or add 2 to the number of teeth, mul- 
whoie Diam- tiply the sum by s and product will be the whole 
diameter. 

l^:/*, or one tenth of thickness of tooth at pitch-line 
Clearance. equals amount added to bottom of space for clearance. 
Some mechanics prefer to make /'equal to -^^ of the 
working dej^th of teeth, or .0625 D''. One-tenth of 
the thickness of tooth at pitch-line is more than one- 
sixteenth of working depth, being .07854 D'. 
Example. Exaiuple.— Wheel 30 teeth, \h" circular pitch. 

^^, ^ P' = 1.5" : then ^ = .75" or thickness of tooth equals f". 

Sizes of Blank -■■ * 

p^rt'L fol-'oelr « = l-5''X.3183 = .4775" = a diameter pitch. (See 

of 30 teetii, \y, tables of tooth parts, pages 64-67). 

i?tch.''*''' '''' D'^30X.4775" = 14.325" = diameter of pitch-circle. 

D==(30x2)X.4775" = 15.280" = diameter of adden- 
dum circle. 

/—jj) of .75'' = .075" = clearance at bottom of sj^ace. 

*D" = 2x.4775" = .9549"rr:working depth of teeth. 

D" + /_i.2x.4775"-f.075" = 1.0299"=whole depth of 
space. 

s+/=.4775"-l-.075" = .5525"=depth of space inside 
of pitch-line. 

D" = 2s or the working depth of teeth is equal to 
two diameter pitches. 

In making calculations it is well to retain the fourth 
place in the decimals, but when drawings are passed 
into the workshop, three places of decimals ai'e suffi- 
cient. 



PROVIDENCE, R. I. 




Fig. 5, Spur Gearing. 



8 BROWN & SHARPE MFG. CO. 

Distance be- The distance between the centers of two wheels is 

of two Gears, evidently equal to the radius of pitch-circle of one wheel 

added to that of the other. The radius of pitch-circle 

is equal to s multiplied by one- half the number of teeth 

in the wheel. 

Hence, if we know the number of teeth in two w^heels, 
in mesh, and the circular pitch, to obtain the distance 
between centers we first find s ; then multiply s by one- 
half the sum of number of teeth in both wheels and the 
product will be distance betw^een centers. 

Example — What is the distance between the centers 
of two wheels 35 and 60 teeth, 1 J" circular pitch. We 
first find s to be IJ'x .3183 = . 3979". Multiplying by 
47.5 (one-half the sum of 35 and 60 teeth) we obtain 
18.899" as the distance between centers. 
ShrSkTge^fn P*lttrrn Gears should be made large enough to 
Gear Castings, allow for shrinkage in casting. In cast iron the shrinkage 
is about ^ inch in one foot. For gears one to two feet 
in diameter it is well enough to add simply j^ of 
diameter of finished gear to the pattern. In gears 
about six inches diameter or less, the moulder will 
generally rap the pattern in the sand enough to make 
any allowance for shrinkage unnecessary. In pattern 
gears the spaces between teeth should be cut wider 
than finished gear spaces to allow for rapping and to 
avoid having too much cleaning to do in order to have 
gears run freely. In cut patterns of iron it is generally 
Metal Pattern enough to make spaces .015" to .02' wider. This 
makes clearance .03" to .04" in the patterns. Some 
moulders might want .06" to .07" clearance. 

Metal patterns should be cut straight ; they work 
better with no draft. It is well to leave about .005" to 
be finished from side of patterns after teeth are cut ; 
this extra stock to be taken away from side where 
cutter comes through so as to take out places w^here 
sto^k is broken out. The finishing should be done 
with file or emery wheel, as turning in a lathe is likely 
to break out stock as badly as a cutter might do. 

If cutters are kept sharp and care is taken when 
coming through the allowance for finishing is not nec- 
essary and the blanks may be finished before they are 
cut. 



Gears. ° 



PROVIDENCE, R. I. 



CHAPTER III. 
SIHGLE-CHRVE GEARS OF 30 TEETH AHD OVER. 



Single-curve teeth are so called because they have ^eeth^^ Curve 
"but one curve by theory, this curve forming both face 
and flank of tooth sides. In any gear of thirty teeth 
and more, this curve can be a single arc of a circle 
whose radius is one-fourth the radius of the pitch 
circle. In gears of thirty teeth and more, a fillet is 
added at bottom of tooth, to make it stronger, equal 
in radius to one-sixth the widest part of tooth space. 

A cutter formed to leave this fillet has the advantage 
of wearing longer than it would if brought up to a 
-corner. 

In gears less than thu*ty teeth this fillet is made the 
same as just given, and sides of teeth are formed with 
moi-e than one arc, as will be shown in Chapter VI. 

Havino* calculated the data of a cfear of 30 teeth, f Example of a 

. . 1 . . ^ Gear, N=30, P' 

inch circular pitch (as we did in Chapter II. for 1^" =\". 
pitch), we proceed as follows : 

1. Draw pitch circle and point it off into parts equal Geometrical 

. 1 IP ii • ^ -i 1 Construction. 

to one-halt the cn*cular pitch. Fig. 6. 

2. From one of these points, as at B, Fig. 6, draw 
radius to pitch circle, and upon this radius describe a 
semicircle ; the diameter of this semicu'cle being equal 
to radius of pitch circle. Draw addendum, working 
depth and whole depth circles. 

3. From the point B, Fig. 6, where semicu'cle, pitch 
€u-cle and outer end of radius to pitch circle meet, lay 
off a distance upon semicircle equal to one-fourth the 
radius of pitch circle, shown in the figure at BA, and 
is laid off as a chord. 

4. Through this new point at A, upon the semicircle, 
draw a circle concentric to pitch circle. This last is 



10 



BEOWN & SHAEPE MFG. CO. 



T^ie;. 






r- 


GEAR, 30 TEETH, 


Q. 


i'CIRCULAR PITCH. 


fe 


P'= for .75" 


CO 


N=30 


ID 

Q 


p =4.1888" 


< 


t = .375' 


S = .2387" 




0°= .4775' 




S+/= .2762' 




D"+/- .5150' 




D'=7.1610' 




D =7.6384" 





SINGLE CURVE GEAR. 



PROVIDENCE, R. I. 



11 



called the base circle, and is the one for centers of 
tooth arcs. In the system of single curve gears, we 
have adopted the diameter of this circle is .968 of the 
diameter of pitch circle. Thus the base circle of any 
gear 1 inch pitch diameter by this system is .968". 
If the pitch circle is 2" the base circle will be 1.936." 

5. With dividers set to one-quarter of the radius of 
pitch circle, draw arcs forming sides of teeth, placing 
one leg of the dividers in the base circle and letting 
the other leg describe an arc through a point in the 
pitch circle that was made in laying off the parts equal 
to one-half the circular pitch. Thus an arc is drawn 
about A as center through B. 

6. With dividers set to one-sixth of the widest part 
of tooth space, draw the fillets for strengthening teeth 
at their roots. These fillet arcs should just touch the 
whole depth circle and the sides of teeth already 
described. 

Single curve or involute gears are the only gears involute^ Gear- 
that can run at varying distance of axes and transmit ^"^• 
unvai'ving angular velocity. This peculiarity makes 
involute gears specially valuable for diiving rolls or 
any rotating pieces, the distance of whose axes is 
likely to be changed. 

The assertion that clears crowd harder on bearings Pressure on 

^ " bearings. 

when of involute than when of other forms of teeth, 
has not been proved in actual practice. 

Before taking next chapter, the learner should make Practice, be- 

11- £ OA J. XI T o *^^*® taking, 

several drawmgs oi gears dO teeth and more, bay next chapter, 
make 35 and 70 teeth 1\" P'. Then make 40 and 65 
teeth I" F. 

An excellent practice will be to make drawing on 
cardboard or Bristol-board and cut teeth to lines, thus 
making paper gears ; or, what is still better, make them 
of sheet metal. By placing these in mesh the learner 
can test the accui'acy of his work. 



12 BBOWN & SHAKPE MFO. CO. 



CHAPTER IV. 

RACK TO MESH WITH SINGLE-CDRVE GEARS HAVIH6 
30 TEETH AND OVER, 



Tuade^ ^re \ra- This gear (Fig. 7) is made precisely the same as gear 
tory to drawing iu Chapter III. It makes no difference in which direc- 
tion the construction radius is drawn, so far as obtain- 
ing form of teeth and making gear are concerned. 

Here the radius is drawn perpendicular to pitch line 
of rack and through one of the tooth sides, B. A semi- 
circle is di-awn on each side of the radius of the pitch 
circle. 

The points A and A' are each distant from the point 
B, equal to one-fourth the radius of pitch circle and 
correspond to the point A in Fig. 6. 

In Fig. 7 add two lines, one passing through B and 

A and one through B and A'. These two lines form 

angles of 75^° (degrees) with radius BO. Lines BA 

and BA' are called lines of pressure. The sides of 

rack teeth are made perpendicular to these lines. 

Hack. A Rack is a straight piece, having teeth to mesh 

with a gear. A rack may be considered as a gear of 

infinitely long radius. The circumference of a circle 

• approaches a straight line as the radius increases, and 

when the radius is infinitely long any finite part of the 

Construction circumference is a straight line. The pitch line of a 

of Pitch Line of . .... 

Hack. rack, then, is merely a straight line just touching the 

pitch circle of a gear meshing with the rack. The 
thickness of teeth, addendum and depth of teeth 
below pitch line are calculated the same as for a wheel. 
(For pitches in common use, see table of tooth parts.) 

A rack to mesh with a single-curve gear of 30 teeth 
or more is drawn as follows : 

1. Draw straight pitch line of rack ; also draw ad- 
Tig. 7. dendum line, working depth line and whole depth line, 

each parallel to the pitch line (see Fig. 7). 



PROVIDENCE, R. I. 



13 




RACK TO MESH WITH SINGLE CURVE GEAR 
HAVING 30 TEETH AND OVER. 



14 BROWN & SHAKPE MFG. CO. 

2. Point off tlie pitch line into parts equal to one- 
half the circular pitch, or = t. 

3. Through these points draw lines at an angle of 
75J° with pitch lines, alternate lines slanting in oppo- 
site directions. The left-hand side of each rack tooth 
is perpendicular to the line BA. The right-hand side 
of each rack tooth is perpendicular to the line BA'. 

4. Add fillets at bottom of teeth equal to \ of the 
width of spaces between the rack teeth at the adden- 
dum line. 

sMes^of^ Rick Tl^® sketch. Fig. 8, will show how to obtain angle of 
Teeth. sides of rack teeth, directly from pitch line of rack, 

without di-awing a gear in mesh with the rack. 




Upon the pitch line b h\ draw any semicircle — 
baa' b'. From point b lay off upon the semicircle 
the distance b a, equal to one-quarter of the diameter 
of semicircle, and di'aw a straight line through b and a. 

This line, b a, makes an angle of 75 J ° with pitch line 
b b', and can be one side of rack tooth. The same 
construction, b' a\ will give the inclination 75j° in the 
opposite direction for the other side of tooth. 

The sketch, Fig. 9, gives the angle of sides of a tool 
for planing out spaces between rack teeth. Upon any 
line OB draw circle OABA'. From B lay off distance 
BA and BA', each equal to one-quarter of diameter of 
the ckcle. 

Draw lines OA and OA'. These two lines form an 
angle of 29°, and are right for inclination of sides of 
rack tool. 



PROVIDENCE, R, I. 



15 



Make end of rack tool .31 of cii'cular pitch, and then width of Rack 



Tool at end. 



round the corners of the tool to leave fillets at the 
bottom of rack teeth. 

Thus, if the circular pitch of a rack is 1^" and we 
multiply by .31, the product .465" will be the width of 
tool at end for rack of this pitch before corners are 
taken off. This width is shown at x y. 




This sketch and the f oreo-oinf^f rule are also ri^ht for worm Thread 

^ ^ ^ Tool. 

a worm-thread tool, but a worm-tlu-ead tool is not 
usually rounded for fillet. In cutting worms, leave 
width of top of thread .335 of the circular pitch. 
"When this is done, the depth of thread will be right. 



16 BEOWN 8l SHAEPE MFG. CO. 



CHAPTER V. 

DIAMETRAL PITCH— SIZING BLAMS AND TEETH— DISTANCE 
BETWEEN CENTERS OF WHEELS. 



In making drawings of gears, and in cutting racks,, 
necessary to it is necGssary to know the circular pitch, both ou 
cuiar Pitch. account of Spacing teeth and calculating theu* strength. 
It would be more convenient to express the cu-cular 
pitch in whole inches, and the most natural divisions 
In a complete of an inch, as 1" P', f" P', -J-" P', and so on. But as 
Pitch ^circum! the cu'cumference of the pitch circle must contain the 
contSrfthe^cir- cii"cular pitch some whole number of times, corre- 
some whofe sponding to the number of teeth in the gear, the 
Smes.^ ® ^ ° ^ diameter of the pitch circle will often be of a size not 
readily measured with a common rule. This is because 
the cii'cumference of a cii'cle is equal to 3.1416 times 
the diameter, or the diameter is equal to the circum- 
ference multiplied by .3183. 

In practice, it is better that the diameter should be 
Pitch, i n of somc size conveniently measured. The same applies 

Terms ot the "^ ■'■■'■ 

Diameter. to the distance between centers. Hence it is generally 
more convenient to assume the pitch in terms of the 
diameter. In Chapter II. Avas given a definition of a 
diameter pitch, and also how to get a diameter pitch 
from the circular pitch. 

We can also assume a diameter pitch and pass to its 
Circular Pitch equivalent circular pitch. If the circumference of the 

a^nd a Diame- ^ ^ 

ter Pitch. pitch circle is divided by the number of teeth m the 

gear, the quotient will be the circular pitch. In the 
same manner, if the diarneter of the pitch circle is 
divided by the number of teeth, the quotient will be a 
diameter pitch. Thus, if a gear is 12 inches pitch 
diameter and has 48 teeth, dividing 12" by 48, the 
quotient J" is a diameter pitch of this gear. In prac- 



PROVIDENCE, R. I. 17 

tice, a diameter pitch is taken in some convenient part 

of an inch, as ^" diameter ])itch, and so on. It Abbreviation 

' ^ , ■"■,' of Diameter 

is convenient in calculation to designate one of these Pitch, 
diameter pitches by s, as in Chapter II. Thus, for ^" 
diameter pitch, s is equal to J". Generally, in speak- 
ing of diameter pitch, the denomiuator of the fraction 
only is named. J" diameter pitch is then called 3 
diametral 2')itch. That is, it has been found more con- 
venient to take the reciprocal of a diameter pitch in 
making calculation. The reciprocal of a number is 1, 
divided by that number. Thus the reciprocal of J is a Numbe?!^ 
4, because \ goes into 1 four times. 

Hence, we come to the common definition : 

Diametral Pitch is the number of teeth to one inch pitcn. 
of diameter of pitch circle. Let this be denoted by P. 
Thus, J" diameter pitch we would call 4 diametral 
pitch or 4 P, because there would be 4 teeth to every 
inch in the diameter of pitch circle. The circular 
jDitch and the different parts of the teeth are derived 
from the diametral pitch as follows. 

^•''1^ ^' = P', or 3.1416 divided by the diametral pitch 
is equal to the circular pitch. Thus to obtain the cir- 
cular for 4 diametral pitch, we divide 3.1410 by 4 and 
get .7854 for the circular pitch, corresponding to 4 ^, ^J^^^.^^jl"] J^^^ 
diametral pitch. JmiPit^h.^""^' 

In this case we would write P=4,P'=:.7854", S=r^". 
^" — s, or one inch divided by the number of teeth to 
an inch, gives distance on diameter of pitch circle 
occupied by one tooth. The addendum or face of 
tooth is the same distance as s. 

|=P, or one inch divided by the distance occupied 
by one tooth equals number of teeth to one inch. 

^•p=t, or 1.57 divided by the diametral pitch sfives Calculations 

•^ "^ ■•■ OforSizesof 

thickness of tooth at pitch line. Thus, thickness of^^^^,"'^^ ^^^ 

' tooth p arts 

teeth along the pitch line for 4 diametral pitch is .392". when the Dia- 

<^ ^ i naetral Pitch is 

p = D', or number of teeth in a gear divided by the^^^' 
diametral pitch equals diameter of the pitch circle. 
Thus for a wheel, GO teeth, 12 P, the diameter of 
pitch circle will be 5 inches. 

-p- =D, or add 2 to the number of teeth in a wheel 
and divide the sum by the diametral pitch, and the 



•en. 



18 BROWN & SHARPE MFG. CO. 

quotient will be the lohole diameter of the gear or the 
diameter of the addendum circle. Thus, for 60 teeth, 
12 P, the diameter of gear blank will be SyV inches. 

D,=P, or number of teeth divided by diameter of 
pitch circle in inches, gives the diametral pitch or 
number of teeth to one inch. Thus, in a wheel, 24 
teeth, 3 inches pitch diameter, the diametral pitch is 8. 

~^-' =P, or add 2 to the number of teeth; divide the 
sum by the whole diameter of gear, and the quotient 
will be the diametral pitch. Thus, for a wheel ^^q" 
diameter, 14 teeth, the diametral pitch is 5. 

P D'=N, or diameter of pitch circle, multiplied by 
diametral pitch equals number of teeth in the gear. 
Thus, in a gear, 5 pitch, 8" pitch diameter, the number 
of teeth is 40. 
A Diameter In futui'e, when we speak of a diameter pitch, we 
shall mean the addendum distance or 6\ If we speak 
of so many diameter pitches, we shall mean so many 
The Diame- times s, (-rr =:S). When we say the diametral pitch we 

tral Pitch. ' 1^ ^ •' ^ i 

shall mean the number of teeth to one inch of diameter 
of pitch circle, or P, (^'=P). 
ametrar^PiSh When the circular pitch is given, to find the corre- 
PitS ^"^^^^^sponding diametral pitch, divide 3.1416 by the circular 
pitch. Thus 1.57 P is the diametral pitch correspond- 
ing to 2-inch circular pitch, {^^}-^ = V). 
Example. What diametral pitch corresponds to \" circular 

pitch ? Kemembering that to divide by a fraction we 
multiply by the denominator and divide by the numer- 
ator, we obtain 6.28 as the quotient of 3.1416 divided by 
\ . 6.28 P, then, is the diametral pitch corresponding 
to ^ circular pitch. This means that in a gear of ^ 
inch circular pitch there are six and twenty-eight one 
hundredths teeth to every inch in the diameter of the 
pitch circle. In the table of tooth parts the diametral 
pitches corresponding to circular pitches are carried 
out to four places of decimals, but in practice two 
places of decimals are enough. 

When two gears are in mesh, so that their pitch 
circles just touch, the distance between then* axes or 
centers is equal to the sum of the radii of the two 



PROVIDENCE, R. I. 19 

gears. The number of the diameter pitches between 

centers is equal to half the sum of number of teeth in 

both gears. This principle is the same as given in 

Chapter II., page 6, but when the diametral pitch and ^^'^^ ^ ^^^ 

numbers of teeth in two gears are given, we add the tweon centers. 

number of teeth together and divide half the sum by 

the diametral pitch. 

A gear of 20 teeth, 4 P, meshes with a gear of 50 Example. 
teeth : what is the distance between their axes or 
centers ? Adding 50 to 20 and dividing half the sum 
by 4, we obtain 8j" as the center distance. 

The term diametral pitch is also applied to a rack. 
Thus, a rack 3 P, means a rack that will mesh with a 
gear of 3 diametral pitch. 

It will be seen that if the expression for a diameter -p?' J ^^^°^^ J 
pitch has any number except 1 for a numerator, we^^^ch. 
cannot express the diametral pitch by naming the 
denominator only. Thus, if the addendum or a diam- 
eter pitch is f ^, the diametral pitch will be 2^, because 
1 divided by -^-^ equals 2^. 



20 BEOWN S: SaAEPE MFG. CO. 



CHAPTER VI. 

SINGLE-CDRVE GEARS HAVIHG LESS THAN 30 TEETH— GEARS AKD 
RACKS TO MESH WITH GEARS HAVING LESS THAN 30 TEETH. 



Construction, j^ pio-, 10, tlie constructioii of the rack is the same 

Fig. 10. . 

as the construction of the rack iu Chapter lY. The 
gear in Fig. 10 is drawn from base circle out to adden- 
dum circle, by the same method as the gear in Chapter 
III., but the sj^aces inside of base circle are drawn as 
follows : 
Gears ^i^^ low ^^ g^ars, 12 and 13 teeth, the sides of spaces 
Teeui^^^^ ^^ inside of base circle are parallel for a distance not 
more than J of a diameter pitch, or ^ s ; gears 14, 
15 and 16 teeth not more than -J- s ; 17 to 20 teeth, 
not more than -J- s. In gears with more than 20 teeth 
the parallel constructioii is omitted. 
Construction Then, witli one leg of dividers in pitch circle in 

of Fig. 10 con- Till. 

tinued. center of next tooth, e, and other leg just touching 

one of the parallel lines at b, continue the tooth side 
into c, until it will touch a fillet arc, whose radius is 
■1- the width of space at the addendum circle. The 
part, b' c\ is an arc from center of tooth g, etc. The 
flanks of teeth or spaces in gear. Fig. 11, are made the 
same as those in Fig. 10. 

This rule is merely conventional or not founded 
upon any principle other than the judgment of the de- 
signer, to effect the object to have spaces as wide as 
practicable, just below or inside of base circle, and. 
then strengthen flank with as large a fillet as will clear 
addenda of any gear. If flanks iu any gear will clear 
addenda of a rack, they will clear addenda of any 

Internal Gear, other gear, except internal gears. An internal gear i& 
one having teeth ujdoii the inner side of a rim or ring. 
Now, it will be seen that the gear. Fig. 10, has teeth 



PROVIDENCE, R. I. 



21 




22 



BROWN & SHARPE MFG. CO. 



too much rounded at the points or at the addendum 

ch'cle. In gears of pitch coarser than 10 to inch (10 

Addend? of ■^)' ^^^ having less than 30 teeth, this rounding 

Teeth. becomes objectionable. This rounding occurs, because 

in these gears arcs of circles depart too far from the 

true involute curve, being so much that points of 

teeth get no bearing on flanks of teeth in other wheels. 

In gear, Fig. 11, the teeth outside of base circle are 

made as nearly true involute as a workman will be able 

to get without special machinery. This is accomplished 

Approxima- ^g f oUows : draw three or four tanp^ents to the base 

tion to True In- o 

volute. cii'cle, i i', Jj\ k Jc', 1 1', letting the points of tangency 

on base cu"cle i\J', ^', I' be about ^ or i the circular pitch 
apart ; the first point, i\ being distant from ^, equal to 
I the radius of pitch circle. With dividers set to J 
the radius of pitch circle, placing one leg in ^', draw 
the arc, a' i j; with one leg in j', and radius j' j, 
drawj k; with one leg in k\ and radius k' k draw k I. 
Should the addendum circle be outside of Z, the tooth 
side can be completed with the last radius, l' I. The 
arcs, a' ij\ j k and k I, together form a very close 
approximation to a true involute from the base circle, 
i' J' k' v. The exact involute for gear teeth is the 
€urve made by the end of a band when unwound from 
a cylinder of the same diameter as base circle. 

The foregoing operation of drawing tooth sides, 
although tedious in description, is very easy of practical 
application. 
Hounding of It will also be seen that the addenda of rack teeth 

Addendaof -in , ,^ i\ ^ , 

Hack. in Fig. 10, interfere with the gear-teeth nanks, as at 

7n n; to avoid this interference, the teeth of rack, Fig. 
11, are rounded at points or addenda. 

It is also necessary to round off the points of invo- 
lute teeth in high numbered gears, when they are to 
interchangewith low-numbered gears. In interchange- 
able sets of gears the lowest-numbered pinion is usual- 
Tempietsly 12. Just how much to round off is best learnt by 

necessary for "^ o»,,i leji- ii 

Rounding off makino" templets of a lew teeth out oi thin metal or 

Points of teeth. -, ° n / ,i i i , 

cardboard, for the gear and rack, or, two gears re- 
quired, and fitting addenda of teeth to clear flanks. 
However accurate we may make a diagram, it is quite 



PROVIDENCE, R. 



23 




24 BROWN & SHARPE MFG. CO. 

as well to make templets in order to shape cutters 
accurately. 

A set of cutters for interchangeable gears will give 

good results in finer pitches, made to Fig.6, in Chapter 

III., for 30 teeth and over, and to Fig. 10, for less 

than 30 teeth, arranged upon the following : 

a ''sir or cm- -^^^' cutting 12 and 13 teeth, make cutter to diagram 

ters. of 14 teeth. 

For 14 to 16 teeth, diagram of 16 teeth. 



17 


" 20 






" 18 


21 


" 25 






" 22 


26 


" 34 






" 28 


35 


" 54 






" 38 


55 


"134 






" 55 


135 


" rack. 






"100 



Cut a rack with cutter made to diagram for 100 
teeth. 

For gears, 10 P and coarser, it is better to make 
cutters to corrected diagrams, as in Fig. 11. When 
corrected diagrams are made, as in Fig. 11, take the 
following : 

For 12 and 13 teeth, diagram of 12 teeth. 



" 14 


to 16 


" 17 


" 20 


" 21 


" 25 


" 26 


" 34 


" 35 


" 54 


" 55 


" 134 



" 14 


u 


u 17 


a 


" 21 


a 


" 26 


u 


" 35 


a 


" 55 


(c 


"135 


« 



" 135 " rack, " " 

Templets for large gears must be fitted to run with 
12 teeth, etc. 

We obtain the form for involute cutters by special 
machinery. 



PEOVIDENCE, R. I, 25 



CHAPTER VII. 
DODBLE-CDRVE TEETH— GEAR, 15 TEETH— RACK. 



In double-curve teeth tlie fonnation of tooth sides Aii Doubie- 

ourve Tooth 

changes at the pitch Hue. In all gears the part of *'aces are con- 
teeth outside of pitch line is convex ; in some gears 
the sides of teeth inside pitch line are convex ; in some, 
radial ; in others, concave. Convex faces and concave 
flanks are most familiar to mechanics. In interchange- 
able sets of gears, one gear in each set, or of each 
pitch, has radial flanks. In the best practice, this gear 
has fifteen teeth. Gears ivith more than fifteen teeth, 
have concave flanks ; gears Tvith less than fifteen teeth, 
have convex flanks. Fifteen teeth is called the Base 
of "this system. 

We will first draw a c^ear of fifteen teeth. This , construction 

° of Fig. 12. 

fifteen-tooth construction enters into gears of any 
number of teeth and also into racks. Let the gear be 
3 P. Having obtained data, we proceed as follows : 

1. Draw pitch circle and point it off into parts equal 
to one-thirtieth of the circumference, or equal to thick- 
ness of tooth =:^. 

2. From the center, through one of these points, as 
at T, Fig. 12, draw line OTA. Draw addendum and 
whole-depth circles. 

3. About this point, T, with same radius as 15-tooth 
pitch circle, describe arcs A K and O k. For any other 
double-curve gear of 3 P., the radius of arcs, A K and 
O ^% will be the same as in this 15-tooth gear=:2^". 
In a 15-tooth gear, the arc, O X;, passes through the 
center O, but for a gear having any other number of 
teeth, this construction arc does not pass through 
center of gear. Of course, the 15- tooth radius of arcs, 
A K and O k, is always taken from the pitch we are 
working with. 



26 



BKOWN & SHARPE MFG. CO. 



P — 3 

N=15 

P'= 1.-0472'' 

if =. .5236'- 

= .3333' 

&"= .6666' 

S+f^ .3857" 

D"4i/= .7190" 

D''^ 5.0000" 

D = 5.6666? 




JBTig. i;: 

DOUBLE CURVE GEAR. 



PROVIDENCE, R. I. 2T 

4. Upon tliese arcs on opposite sides of lines OTA, 
lay off tooth thickness, A K and O /.:, and draw line 

5. Perpendicular to K T k, draw line of pressure, 
L T P ; also through O and A, draw lines A R and O r, 
perpendicular to K T 7c. The line of pressure is at 
an angle of 78° with the radius of gear. 

6. From O, draw a line O R to intersection of A R 
with K T k. Through point c, where O R intersects 
L P, describe a circle about the center, O. In this 
circle one leg of dividers is placed to describe tooth 
faces 

7. The radius, c d, of arc of tooth faces is the 
straight distance from c to tooth-thickness point, h, 
on the other side of radius, O T. With this radius, c b, 
describe both sides of tooth faces. 

8. Draw flanks of all teeth radial, as Oe and Of 
The base gear^ 15 teeth only, has radial flanks. 

9. With radius equal to one sixth of the widest part 
of space, as g A, draw fillets at bottom of teeth. 

The foreffoinof is a close approximation to epicy- . Approxima- 

^ " ^^ ^ "^ tion to Epicy- 

cloidal teeth. To get exact teeth, make two 1 5- tooth cioidai Teeth, 
gears of thin metal. Make addenda long enough to 
come to a point, as at n and q. Make radial flanks, as 
at m and p, deep enough to clear addenda wdien geai's 
are in mesh. First finish the flanks, then fit the long 
addenda to the flanks when gears are in mesh. 

When these two templet gears are alike, the centers ^t an aard 
are the right distance apart and the teeth interlock 
without backlash, they are exact. One of these tem- 
plet gears can now be used to test any other templet 
gear of the same pitch. 

Gears and racks will be right when they run cor- 
rectly with one of these 1 5-tooth templet gears. Five 
or six teeth are enough to make in a gear templet. 

Double CURVE Rack. — Let us draw a rack 3 P. ^ ^?^^4?-''^7® 

_^ _ _ KacK, r ig. lo. 

Having obtained data of teeth we proceed as follows : 

1. Draw pitch line and point it off in parts equal 
to one half the circular pitch. Draw addendum and 
whole-depth lines. 

2. Through one of the points, as at T, Fig. 13, draw 
line OTA perpendicular to pitch line of rack. 



28 



BROWN & SHARPE MFG. CO. 




DOUBLE CURVE RACK. 



PROVIDENCE, R. I. 



29 



3. About T make precisely the same construction as 
ivas made about T in Fig. 12. That is, with radius of 
1 5-tooth pitch circle and center T draw arcs O k and 
A K ; make O k and A K equal to tooth thickness ; 
draw K T k ; draw O r, A R, and line of pressui-e, each 
perpendicular to K T ^. 

4. Through R and r, draw lines parallel to O A. 
Through intersections c and c' of these lines, with 
pressure line L P, draw lines parallel to pitch line. 

5. In these last lines place leg of dividers, and draw 
faces and flanks of teeth as in sketch. 

6. The radius c d' of rack- tooth faces is the same 
length as radius c d of rack-tooth flanks, and is the 
straight distance from c to tooth-thickness point h on 
opposite side of line O A. 

7. The radius for fillet at bottom of rack teeth is 
equal to \ of the widest part of tooth sj)ace. This 
radius can be varied to suit the judgment of the 
designer, so long as a fillet does not interfere with 
teeth of engaging geai*. 




Fis- T-4. 



Racks of the same pitch, to mesh with interchange- 
able gears, should be alike when placed side by side, 
and fit each other when placed together as in Fig. 14. 

In Fig. 13, a few teeth of a 15-tooth wheel are shown 
in mesh with the rack. 



30 BROWN & SHARPE MFG. CO. 



CHAPTER VIII. 

DODBLE-CDRYE GEARS, HAYING MORE AND LESS THAN 
15 TEETH— ANNULAR GEARS. 



of Sgf™'*''' Let us di-aw two gears, 12 and 24 teeth, 4 P, in 
mesh. In Fig. 15 the construction lines of the lower 
or 24-tooth gear are full. The upper or 12-tooth gear 
construction lines are dotted. The line of pressure, 
L P, and the line K T ^ answer for both gears. The 
arcs A K and O k are described about T. The radius 
of these arcs is the radius of pitch cii'cle of a gear 15 
teeth 4 pitch. The length of arcs A K and O h is the 
tooth thickness for 4 P. The line K T A; is obtained 
the same as in Chapter VII. for all double-curve gears, 
the distances only varying as the pitch. Having drawn 
the pitch circles, the line K T ^, and, perpendicular to 
K T ^% the lines A K, O r and the line of pressure 
L T P, we proceed with the 24-tooth gear as follows : 

1. From center C, through r, draw line intersecting 
line of pressure in m. Also draw line from center C 
to R, crossing the line of pressure L P at c. 

2. Through ni describe circle concentric with pitch 
circle about C. This is the circle in which to place 
one leg of dividers to describe flanks of teeth. 

3. The radius, m n, of flanks is the straight distance 
from m to the first tooth-thickness point on other side 
of line of centers, C C', at v. The arc is continued to 
^^, to show how constructed. This method of obtain- 
ing radius of double-curve tooth flanks applies to all 
gears having more than fifteen teeth. 

4. The construr tion of tooth faces is similar to 15- 
tooth wheel in Chapter YII. That is : Draw a circle 
through c concentiic to pitch circle ; in this circle 
place one leg of dividers to di'aw tooth faces, the 
radius of tooth faces being c h. 



PROVIDENCE, R. I. 



31 




P=4 

N=12 and 24 
P'= .7854" 
t = .3927" 
S = .2500" 
D-= .5000' 
St/ = .2893° 
0"+/ = .5393' 



(D'^3" 
PINION (0 = 3^' 

(D'=6' 



DOUBLE CURVE GEARS IN MESH 



32 BROWN & SHAEPE MFG. CO. 

of ?i'^'2on^ ^- The railius of fillets at roots of teeth is equal to 
tinued. one-sixth the width of space at addendum circle. 

Flanks for 12, "^^^ coiistructious foF flaiiks of 12, 13 and 14 
13 and 14 Teeth, teeth are similar to each other and as follows : 

1. Through center, C, draw line from E, intersecting 
line of pressure in u. Through u draw circle about 
C. In this circle one leg of dividers is placed for 
drawing flanks. 

2. The radius of flanks is the distance from u to 
the first tooth-thickness point, e, on the s(une side of 
C T C. This gives convex flanks. The arc is con- 
tinued to V, to show construction. 

3. This arc for flanks is continued in or toward the 
center, only about one sixth of the working depth (or 
J s.) ; the lower part of flank is similar to flanks of 
gear in Chapter VI. 

4. The faces are similar to those in 15-tooth gear, 
Chapter VII., and to the 24:-tooth gear in the fore- 
going, the radius being lo y ; the arc is continued to a*, 
to show construction. 

AnnuiarGears. Annular Gears. Gears with teeth inside of a rim 
or ring are called Annular or Internal Gears. The 
construction of tooth outlines is similar to the fore- 
going, but the spaces of a spur external gear become 
the teeth of an annular gear. 

Prof. MacCord has shown that in the system just 
described, the pinion meshing with an annular gear, 
must diff'er from it by at least fifteen teeth. Thus, 
a gear of 24 teeth cannot work with an annular gear 
of 36 teeth, but it will work with annular gears of 39 
teeth and more. An annular gear differing from its 
mate by less than 15 teeth can be made. This will be 
shown in Part II. 

Annu'ar-gear patterns require more clearance f< 
moulding than external or spur gears. 
Pinions. In speaking of different-sized gears, the smallest 

ones are often called "pinions." 

The angle of pressure in all gears except involute, 
constantly changes. 78° is the pressure angle in 
double-curve, or epicycloidal gears for an instant 
only ; in our example, it is 78° when one side of a 



PROVIDENCE, R. I. 33^ 

tooth reaches the line of centers, and the pressure 
against teeth is applied in the direction of the arrows. 

The pressure angle of involute gears does not 
change. An explanation of the term angle of j)ressure 
is given in Part II. 

We obtain the forms for epicycloidal gear cutters 
by means of a machine called the Odontom Engine. 
This machine will cut original gears with theoretical 
accuracy. 

It has been thoucfht best to make 24 cfear cutters 24 ooubie- 

° ° curve {; ear 

for each pitch. This enables us to fill any require- <^' u tiers for 

^ c/ 1 each Pitch. 

ment of gear-cutting very closely, as the range covered 
by any one cutter is so small that it is exceedingly near 
to the exact shape of all gears so covered. 

Of course, a cutter can be exactly/ right for only one 
gear. Special cutters can be made, if desu'ed. 



34 BROWN & SHABPE MFG. CO. 



CHAPTER IX. 
BEVEL-GEAR BLANKS. 



The i^itcli of bevel gears is always figured at the 
largest pitch diameter. 

Most bevel gears connect shafts that are at right 

angles with each other, and unless stated otherwise, 

we always understand they are so wanted. 

D?a?rIm,''''FiS* The directions given in the following, apply to any 

ft r uTu f n°Sf a^g^® (i^ ^^^ sketch. Fig. 16, the axes are at right 

Jjl^^g^"^®^^ angles). Having decided upon the pitch, numbers of 

teeth and angle of shafts : 

1. Draw center lines of shafts, A O B and COD, 
at angle required. 

2. Parallel to A O B, draw lines a h and c c?, each 
distant from A O B, equal to half the largest pitch 
diameter of one gear. For 24 teeth, 4 pitch, this half 
largest pitch diameter is 3". 

3. Parallel to C O D, draw lines e/and g h, distant 
from COD, equal to half the largest pitch diameter 
of other gear. For gear, 12 teeth, 4 pitch, this half 
largest pitch diameter is 1 J". 

4. At intersections of these four lines, draw lines 
O ^, O^*, O k, and 1/ these lines give the size and 
shajDC of pitch cones. We call them " Cone Pitch 
Lines." 

5. Perpendicular to cone-pitch lines and through 
intersection of lines a b, c d, ef, g h, draw lines m n, 
op, q r. We have also drawn u v to show that another 
gear can be drawn from same diagram. When shafts 
are at right angles, four gears, two of each size, can 
be drawn from same diagram. 

6. Upon lines m n, op, q r, the addenda and depth 
of teeth are laid off, these lines being passed through 
the largest pitch diameter of gears. Lay off the adden- 



PROVIDENCE, R. I. 



35 




33 BROWN & SHARPE MTG. CO. 

dum, it being in these gears ^". This gives distance 
m n, o p, (J r and it v equal to the working depth of 
teeth, in these gears ^". The addendum, of course, is 
measured from cone pitch lines. 

7. Draw lines O m, O n, O o, O p, O q, O r. These 
lines give the height of tseth above cone-pitch lines 
and the working depth of teeth. The teeth become 
smaller as they approach O and would vanish entirely 
at O. It is quite as well never to have length of teeth, 
or face, vi 771' longer than one-third the distance 77i O, 
nor more than 2 J- times the circular pitch. 

8. Having decided upon the length of face, draw 
limiting lines 711' 91 perpendicular to I O, q r' perpen- 
dicular to k O, and so on. 

We have now the outline of section of gears through 

their axes. The distance m r is the v:hole diameter of 

pinion. The distance q o i^i the tohole diameter of 

gear. In practice these diameters are obtained by 

The Whole ^g(/5,^;.^';;^ the drawin"-. The diameter of pinion is 

Diameter of '^ ^ ^ 

Bevel G ear 3,45 ' and of fifcar 6.22". We also find anoles by meas- 

Blanks is gen- '^ o ^ 

eraiiy obtained uriiig drawing with a protractor. In the absence of a 

by Measuring o o j. 

Drawings. j^i'^tractor templates can be cut to drawing. The 
angle formed by line 711 tu' with a h is the angle of 
face of pinion, in this pinion 59° 11' or 59^° nearly. The 
lines q q and g h give us angle of face of gear, for this 
gear 22° 19' or 22J° nearly. The angle formed by 7n n 
with a Z> is called the angle of edge of pinion, in oui' 
sketch 26° 34', or about 261°. The angle of edge of 
gear, line q r with g h is 63° 26', or about 63^°. In 
turning blanks to these angles we place one arm of 
the protractor or template against end of huh, when 
trying angles of a blank. Some designers give angles 
from the axes of gears, but it is not convenient to try 
blanks in this way. The method we have given also 
comes right for angles as figured in compound rests. 

When axes are at right angles, the sum of angles of 
edge in the two gears equals 90°, and the sums of 
angle of edge and face in each gear are alike. 

The angle of axes remaining the same, all pairs of 
•bevel gears of same ratio have the same angle of edge ; 
all pau-s of same ratio and of same numbers of teeth 



PROVIDENCE, R. I. 87 

have the same angles of both edges and faces inde- 
pendent of the pitch. Thus, in all joairs of bevel gears 
having one gear twice as large as the other, with axes 
at right angles, the angle of edge of large gear is C3* 
26', and the angle of edge of small gear is 26° 34'. 

In all pairs of bevel gears with axes at right angles, 
one gear having 24 teeth and the other gear having 12 
teeth, the angle of face of large gear is 22° 19', and the 
angle of face of small gear is 59° 11'. 

The following method of obtaining the whole 
diameter of bevel gears is sometimes preferred: 

From k lay off, upon the cone-pitch line, a distance Another 

•^ ' -^ ^ ' method of ob- 

K w, equal to ten times the w^orkmg depth of the tainiug whoie 

teeth=10D". Now" add j\ of the shortest distance of Blanks. 

w from the line g A, which is the perpendicular dotted 

line 10 it', to the outside pitch diameter of gear, and the 

sum will be the whole diameter of gear. In the same 

manner j\ of w y, added to the outside pitch diameter 

of pinion, gives the w^hole diameter of pinion. 



38 BROWN & SHARPE MFG. CO. 



CHAPTER X. 
BEVEL GEARS— FORM AND SIZE OF TEETH— CDTTIHG TEETH. 



To obtain data for teetli, we need only make di'awing 
of section of one-half of eacli gear. 

1. AVe first draw center lines A O and B O, Fig. 17, 
and the lines g h and c d, then gear blank lines as in 
Chapter IX. 

Shape of teeth Tq obtain shape of teeth in bevel i^ears we do not lay 

In bevel gears. •'• o ./ 

them off on pitch circles the same as we would in spur 
gears. A line running from a point on cone -pitch line 
to center line of a bevel gear and perpendicular to the 
cone-pitch line, is the radius for circle apon which to 
draw outlines of teeth at this point. 

2. Thus, A c is the geometrical pitch circle radius 
for large end of teeth, and A' c' the geometrical pitch 
radius for small end of teeth of wheel. To avoid con- 
fusion, we have transferred the distance A' c' to the 
line A c". 

3. For the pinion we have the geometrical pitch 
cu'cle radius B c for large end of teeth, and the radius 
B' c' for small end of teeth. We have transferred the 
distance B' c' to the line B c'". 

4. About A di'aw the arc c n m, and upon it lay off 
spaces equal to the thickness of tooth at pitch line, 
and di'aw outlines of teeth as in previous chapters. 
That is, for single-curve teeth draw a semicircle upon 
radius Ac and proceed as in Chapter III. For all bevel 
gears cut with a rotary disk-cutter (a common gear- 
cutter), single-curve teeth are preferable. Double- 
curve teeth can be drawn from instructions in Chapters 
VII. and VIII. We now have the shape of teeth at 
large end of gear. Kepeat this operation with radius 
B c about B, and we have form of teeth at large end 
of pinion. 



PROVIDENCE, R. I. 



39 




D"= .266" 

s't/ = -les- 

D"+jr =-.298" 



D"+/ 



Fig. IT 

BEVEL GEARS, FORM AND SIZE OF TEETH 



40 BROWN & SHARPE MFG. CO. 

5. Upon arc of radius A' c' we get shape of teeth at 
small end of gear, and upon arc of radius B' c' we get 
shape of teeth at small end of pinion. 
Sizes of teeth 6. The sizes of tooth parts at small end can be taken 

at small end , ^ 

by diagram, directly from diagram, or they can be calculated. By 
'measuring the distance p' c' and c' o',we obtain adden- 
dum and w^orking depth of teeth at inside of the gears. 
The thickness of teeth at inside is given at 7n n' by 
lines "tn n running to A, the distance m n being the 
thickness of tooth at outside. We designate tooth 
parts at inside by adding one more accent to the 
corresponding designation for tooth parts at outside, 
Avhich are the same as those for spur gears. The tooth 
parts inside are denoted thus: t\ s', D'", s' + f, T>"' -i-f. 
All elements of the teeth run to the point O, where 
axes of gears meet. The sizes of tooth parts at small 
end can be calculated as follows : (These sizes bear the 
same ratio to the corresponding sizes at outside of 
jg'ears that the distance O c' does to O c.) 
Sizes of teeth Dividing the distance O c, which in our example is 

caSation. ^ 2", by O c, which in this case is 3", we obtain f or .666 as 
the ratio. Multiplying the outside sizes by .666, we ob- 
tain the corresponding inside sizes (see Fig. 17). Thus, 
the thickness of teeth at outside being .314, we take ^ of 
.314" and obtain .209" as the thickness of teeth inside. 
Cutting Bev- When cutting bevel gears with rotary cutters the 

notary Cutters, angle of cutter-head is set the same as angle of 
working depth ; thus : To cut the gear we have the 
cutter travel in the direction of line O 79. The angle 
A O^^ is called the " Cutting Angle," being measured 
from the axis of gear. In the method here adopted 
the angle of face of pinion is the same as cutting angle 
of gear, and face angle of gear is the cutting angle of 
pinion ; so that /*, or clearance at bottom of spaces, is 
the same at inside of gear as f at the outside of gear. 
Side Angle The cutter-liead in some gear-cutting engines can 
* be set at an angle sidewise as well as toward the 
work spindle. 

In such engines, for cutting bevel gears, take cutter 
of right curve for teeth about one- third the length of 
tooth from large end ; this gives teeth with sides not 



PROVIDENCE, R. I. 41 

'cnongli curved at small end. In pinions it is some- 
times necessary to lile teeth at small ends ; the cutter 
should, of course, be thin enough to go through space 
at small end. In these engines the spaces at large end 
of teeth are made wide enough by setting the cutter- 
liead sidewise, two settings and two cuts being applied 
to each gear ; but after the first gear the setting is 
•changed once only for each gear. In most engines Different Cut- 
there is no provision for side angle, and the width of chines without 
sjmces is obtained by turning the work spindle and' 
setting cutter out of center. When the cutting angle 
•of gears is greater than 45 '^ the teeth cut by the two 
engines are about alike ; but when the cutting angle is 
less than 45', the same cutter will make the adden- 
dum of teeth more rounding in the cutting engine with 
no side angle than it will in the cutting engine with 
side angle. In such cases a corresponding allowance 

should be made in the shaping of cutters. In orderin<Tf ^,V}\ {"•' ^ ^' ^ a i 

i- c) o M 1 1 1 1 n g Ma- 

bevel-gear cutters it should be stated which kind of «^"ies can be 

'^ usedboth 

engine the cutters are to be used in. The Universal ways. 
Milling Machine can be used for either side angle or 
rotation of work spindle. 

We can plane bevel gears ui) to 16" diameter, theo- We can plane 

,. ,, . , , . . . , Accurate Bevel 

Tetically exact, each gear being an original one. Gears. 



42 BROWN & SHARPE MFG. CO. 



CHAPTER XI. 
WORM WHEELS— SIZING BLAliKS OF 32 TEETH AHD O^ER. 



Worm. ^ WORM is a screw made to mesli with the teeth of 

a wheel called a worm-ioheel. As implied at the end of 
Chapter IV., a section of a worm through its axis is, in 
outline, the same as a rack of corresponding pitch. 
This outline can be made either to mesh with single or 
double curve gear teeth ; but worms are usually made 
for single curve, because, the sides of involute rack 
teeth being straight (see Chapter IV.), the tool for 
cutting worm-thread is more easily made. The thread- 
tool is not usually rounded for giving fillets at bottom 
of worm-thread. 

The rules for circular pitch apj)ly in the size of tooth 
parts and diameter of pitch-circle of worm-wheel. 

Pitch of Worm. The pitch of a worm or screw is usually given in a 
way different from the pitch of a gear, viz. : in number 
of threads to one inch of the length of the worm or 
screw. Thus, if we say a worm is 2 pitch we mean 2 
threads to the inch, or the worm makes two turns to 
advance the thread one inch. But a worm may be 
double-threaded, triple-threaded, and so on. 

To avoid misunderstanding it is better always to 

Worm^Tiirei(f. Call the advance of the worm thread the lead. Thus, a 
worm- thread that advances one inch in one turn we 
call one -inch lead in one turn. A single-thread worm 
4 to 1" is J" lead. We apply the term pitch to the actual 
distance between the threads or teeth, as in previous 
chapters. In single- thread worms the lead and the 
pitch are alike. If we have to make a worm and wheel so- 
many threads to one inch, we first divide 1" hy the num- 
ber of threads to one inch, and the quotient gives us 
the circular pitch. Hence, the wheel in Fig. 19 is ^" 

Linear Pitch, circular pitch. The term linear pitch expresses ex- 



PROVIDENCE, R. I. 



43 




FIG. 18-WORM AND WORM-WHEEL 

The thread of ^A'■opm is left-handed ; Worm is single-threaded. 



u 



BROWN & SHAEP£ MFG. CO. 




IMIOVIDENCE, R. I. 45 

■actly what is meant by circular pitch. Linear pitch 
has the advantage of being an exact use of language 
when ap2:)lied to worms and racks. The number of 
threads to one inch linear, is the reciprocal of the linear 
pitch. 

Multiply 3.141G by the number of threads to one 
inch, and the product will be the diametral pitch of the 
worm-wheel. Thus, we would say of a double-thread 
worm advancing 1" in 1^ turns that: 

Lead= J" or .75". Linear pitch or P'=#" or .375". Drawing of 

•* ■■■ ° W o r m a u d 

Diametral pitch or P = 8.377. See table of tooth parts. Worm-wheei. 
To make drawing of worm and wheel we obtain 
data as in circular pitch. 

1. Draw center line A O and upon it space off the 
distance a h equal to the diameter of pitch-circle. 

2. On each side of these two points lay off the dis- 
tance 5, or the usual addendum = J, , as ^ c and h d. 

3. From c lay off the distance c O equal to the 
radius of the worm. The diameter of a worm is gen- 
erally four or live tim.es the circular pitch. 

4. Lay off the distances c g and d e each equal to /', 
or the usual clearance at bottom of tooth space. 

5. Through c and e draw circles about O. These 
represent the whole diameter of w^orm and the diam- 
eter at bottom of worm-thread. 

6. Draw h O and i O at an angle of 30° to 45° with 
A O. These lines give width of face of worm-wheel. 

7. Through g and d draw arcs about O, ending in 
h O and i O. 

This operation repeated at a completes the outline 
of worm-wheel. For 32 teeth and more, the addendum 
diameter, or D, should be taken at the throat or 
smallest diameter of wheel, as in Fig. 19. Measure 
sketch for whole diameter of wheel-blank. 

The foregoing instructions and sketch are for cases t e e t h o f 
where the teeth of the wheels are finished with a hob. ishea with iiob. 

A HOB is shown in Fig. 20, being a steel piece Hob. 
threaded with the same tool that threads the worm, 
then grooved to make teeth for cutting, and hardened. 

The whole diameter of hob should be 2/, or twice proportionsof 



the clearance larger than the worm. The outer cor- 



Hob. 



46 BROWN & SHARPE MFG. CO. 

ners of hob-thread can be rounded down as far ai the- 
clearance distance. The width at top of the hob-thread 
before rounding should be .-SI of the linear, or circular 
pitch=.31P'. The whole depth of thread should be 
the ordinary working depth jdIus twice the clearance == 
D" + 2/. The diameter at bottom of hob-thread should 
be the same as diameter at bottom of worm-thread.. 
For thread-tool and worm-thread see end of Chapter 
IV. The thickness of cutter for grooving hob can 
be about i the width of thread at top plus J" = 
•^MZ'-f ^". The width of lands at the bottom can be 



FIG, 20.-H0B. 

Right-handed thread. 

about the depth of thread plus i" = D" + 2/-h4". The 
grooves are usually cut with a round-edge cutter, 
the parallel part of cutter just reaching the bot- 
tom of thread, making the half-round bottom of 
grooves below the bottom of thread. In most 
hobs the teeth are not relieved between the grooves, 
because it takes considerable time to file or cut 
the relief. The bur thrown out by the grooving 
cutter can be left on if it is not too rough. The bur 
will make the hob cut faster. Alternate grooves can 
be cut in opposite direction, so that worm will not cut 



PROVIDENCE, R. I. • 47 

faster on one side than on the other. If many wheels 
are to be finished with the hob, the teeth should be 
relieved. 

The teeth of the wheel are first cut as nearly to 
the finished form as practicable ; the hob and worm- 
wheel are mounted ujDon shafts and hob placed in mesh 
as in Fig. 18. The hob is now made to rotate, and is ^j^^^jf^^ ^° "^® 
dropped deeper into the wheel at each revolution of the 
wheel until teeth are finished. The hob generally 
drives the worm-wheel during this o|)eration. The 
Universal Millino- Machine is very convenient for doings ijnive rs a i 

*^ '' ^Milling Ma- 

this work, and with it the distance between axes of chine used in 

,-,,, Hobbing, 

worm and wheel can be readily noted. Where a great 
many wheels are manufactured special machines have 
been made for driving wheels independent of hob. The why a Wheel 
object of hobbing a wheel is to get more bearing sur- 
face of the teeth upon worm -thread. The worm-wheels, 
Figs. 18 and 26, were bobbed. By hobbing we pro- 
duce outline of teeth something like the thread of a nut. 

If we make the diameter of a worm-wheel blank, that worm-wheei 

Blanks with 

is to have less than 30 teeth, by the common rules Less than 30 

Teeth. 

for sizing blanks, and finish the teeth with a hob, we 
shall find the flanks of teeth near the bottom to be un- 
dercut or hollowiuo-. This is caused by the interf er- ^ interference 

*=> "^ of Thread and 

ence spoken of in Chapter YI. Thirty teeth was there Flank, 
^iven as a limit, which Avill be right when teeth are 
made to circle arcs. With pressure angle 75^°, and 
rack-teeth with usual addendum, this interference of 
rack-teeth with flanks of gear-teeth commences at 31 
teeth (31y'\j- geometrically], and interferes with nearly 
the whole flank in wheel of 12 teeth. 

In Fig 21 the blank for worm-wheel of 12 teeth was Fig. 21. 
sized by the same rule as given for Fig. 19. The wheel 
and worai are sectioned to show shape of teeth at the 
mid-plane of wheel. The flanks of teeth are undercut 
by the hob. The worm-thread does not have a good 
bearing on flanks inside of A, the bearing being that of 
a corner against a surface. 

In Fig 22 the blank for wheel was sized so that pitch- ^^^- ^2- 
circle comes midway between outermost part of teeth 
and innermost point obtained by worm-thread. 



48 



BROWN & SHARPE MFG. CO. 



^N^C^fCt^ 




Fis. ^1. 



PROVIDENCE, R. I. 



49 




Fig. '^tS. 



so BROWN & SHAEPE MFG. CO. 

This rule for sizing worm-wheel blanks has been in 
use to some extent. The hob has cut away flanks of 
teeth still more than in Fig. 21. The pitch-circle in 
Fig. 22 is the same diameter as the pitch-circle in Fig. 
21. The same hob was used for both wheels. The 
flanks in this wheel are so much undercut as to mate- 
rially lessen the bearing surface of teeth and worm- 
thread. 
Interference In Chapter VI. the interference of teeth in high- 
numbered gears and racks with flanks of 12 teeth was 
remedied by rounding off the addenda. Although it 
would be more systematic to round off the threads of 
a worm, making them, like rack-teeth, to mesh with 
interchangeable gears, yet this has not generally been 
done, because it is easier to make a worm-thread tool 
with straight sides. 

Instead of cutting away the addenda of worm- 
thread, we can avoid the interference with flanks of 
wheels having less than 30 teeth by making wheel 
blanks larger. 

Tig. 23. The flanks of wheel in Fig. 23 are not undercut, be- 

cause the diameter of wheel is so large that there is 
hardly any tooth inside the pitch-circle. The 
pitch-circle in Fig. 23 is the same size as pitch- 
circles in Figs. 21 and 22. This wheel was sized 
Diameter at by the following rule : Multiply ihepitch diameter of 

Interference, the wheel by .937, and add to the product four times 
the addendum (-l s) ; the sum will be the diameter for 
the blank at the throat or small part. To get the 
whole diameter, make a sketch with diameter of throat 
to the foregoing rule and measure the sketch. 

It is impractical to hob a wheel of 12 to about 16 or 
18 teeth when blank is sized by this rule, unless the 
wheel is driven by independent mechanism and not by 
the hob. The diameter across the outermost parts of 
teeth, as at A B, is considerably less than the largest 
diameter of wheel before it was hobbed. 

In general it is well to size all blanks, as by page 45 
and Figs. 19 and 21, when the wheels are to be 
hobbed. Of course, if the wheel is to be hobbed the 



PROVIDENCE, R. I. 



51 



^\'5XH C/R 




Fig. 33. 



52 BROWN & SHARPE MFG. CO. 

cutter should be thin enough to leave stock for finish- 
ing. The sjDaces can be cut the full depth, the cutter 
being dropped in. 

To get angle of worm-thread, it is best to apply pro- 
tractor directly to the thread, as computing the 
angle affords but little help. Set gear cutter-head as 
near the angle as can be seen from trial with 
protractor upon thread ; cut a few teeth ; try in worm. 
Generally the cutter-head has to be changed before 
the worm will take the right position. 

When worm-wheels are not hobbed it is better to 
s "u^r Wheei^ * *^^^ blanks like a spur-wheel. Little is gained by 
having wheels curved to fit worm unless teeth are fin- 
ished with a hob. The teeth can be cut in a straight 
path diagonally across face of blank, to fit angle of 
worm-thread, as in Figs. 24 and 27. 
Wheels for For dividing wheels to gear-cutting engines the 
Machines. blanks are turned like a spur-wheel and a cutter about 
-^j" larger diameter than the worm, is dropped in, as 
in Figs. 25 and 28, and the worm-thread is slightly 
rounded at the outer corners. The radius for rounding 
thread can be J- the width of thread at the top. 

Some mechanics prefer to make dividing wheels in 
two parts, joined in a plane perpendicular to axis, hob 
teeth ; then turn one part round upon the other, match 
teeth and fasten parts together in the new position,, 
and hob again with a view to eliminate errors. 

With an accurate cutting engine we have found 
wheels like Figs. 25 and 28, not hobbed, every way 
satisfactory. Dividing wheels of 2 feet diameter and 
less are generally made without arms, the part between 
hub and rim being a solid web. As to the different 
Figures 26, 27 wheels, Figs. 26, 27 and 28, when worm is in right 
position at the start, the life-time of Fig. 26, under 
heavy and continuous work, will be the longest. 

Fig. 27 can be run in mesh with a gear or a rack as 
well as with a worm. At the start the contact of teeth 
in this wheel uj^on worm-thread is in points only ; yet 
such wheels have been many years successfully used 
in elevators. 

Fig. 28 is a neat-looking wheel. In gear-cutting 



PROVIDENCE, R. I. 



53 




Fig, 24. 

"Wopm-wheel with teeth cut in a straight path diagonally across face. 
Worna is double-threaded. 



64 



BROWN & SHARPE MFG. CO. 




Fig, 25. 
"Worm and. "Worm- Wheel, for Gear-cutting Engine. 



PKOVIDENCE, R. I. 



65 



Fig. ;^6. 



FiS' 27. 



Fig. 38. 



56 BROWN & SHARPE MFG. CO. 

engines where tlie workman has occasion to turn the 
work spindle by hand, it is not so rough to take hold 
of as Figs. 26 and 27. The teeth are less liable to in- 
jury than the teeth of Figs. 26 and 27. 

Some designers prefer to take off the outermost part 
of teeth in wheels (Figs. 18 and 26), as shown in these 
two figures, and not leave them sharp, as in Fig. 19. 

We do not know that this serves any purpose except 
a matter of looks. 



PROVIDENCE, R. T. 57 



CHAPTER XII. 

SIZING GEARS WHEN THE DISTANCE BETWEEN CENTERS AND THE 
RATIOS OF SPEEDS ARE FIXED-GENERAL REMARKS-WIDTH 
OF FACE OF SPUR GEARS— SPEED OF GEAR CUTTERS— TABLE 
OF TOOTH PARTS, 



Let US suppose that we have two shafts 14" apart, 
center to center, and wish to connect them bv Cfears so center dis- 

' "^ ^ tance and Ratio 

that they will Lave speed ratio 6 to 1. We add the 6fi^e<i- 
and 1 together, and divide 14" by the sum and get 2" 
for a quotient; this 2", multiplied by G, gives us the 
radius of pitch circle of large wheel = 12". In the same 
manner we get 2" as radius of pitch circle of small wheel. 
Doubling the radius of each gear, we obtain 24" and 4" 
as the pitch diameters of the two wheels. The two num- 
bers that form a ratio are called the terms of the ratio, 
AVe have now the rule for obtaining pitch-circle diame- 
ter of two wheels of a given ratio to connect shafts a 
_given distance apart: 

Divide the center distance In/ the sum of the terms of ^"i© for Di- 

^ '^ J ameter of Pitch. 

the ratio; find the product of tin ice the quotient by each *^^^^^^»- 
term separately, and the two products loill le the pitch 
diameters of the two icheels. 

It is well to give special attention to learning the 
rules for sizing blanks and teeth ; these are much 
oftener needed than the method of forming tooth out- 
lines. 

A blank l.j" diameter is to have 16 teeth: what will 
the pitch be ? What will be the diameter of the pitch 
circle ? See Chapter Y. 

A good practice will be to compute a table of tooth 
parts. The work can be compared with the tables 
pages 64, 67. 



68 BROWN & SHARPE MFG. CO. 

In computing it is well to take it to more than four 
places, 7t to nine places = 3.] 41592653. ^ to nine 
places = .318309886. 

There is no such thing as pure rolling contact in 
teeth of wheels ; they always rub, and, in time, will 
wear themselves out of shape and may become noisy. 

Bevel gears, when correctly formed, run smoother 
than spur gears of same diameter and pitch, because 
the teeth continue in contact longer than the teeth of 
spur gears. For this reason annular gears run smoother 
than either bevel or spur gears. 

Sometimes gears have to be cut a little deej^er than 
designed, in order to run easily on their shafts. If 
any departure is made in ratio of pitch diameters it is 
better to have the driving gear the larger, that is, cut 
the follower smaller. For wheels coaiser than eight 
diametral pitch (8 P), it is generally better to cut twice 
around, when accui^ate work is wanted, also for large 
wheels, as the expansion of parts from heat often causes 
inaccurate work when cut but once around. There is 
not so much trouble from heat in plain or web gears as 
in arm gears. 
Width of Spur The width of cast-ii'on gear faces for general pur- 
poses can be made to the following rule : 

Divide 8 hy the diametral 2ntch and add ^" to the 
quotient/ the sum will he vyidth of face for the pitch 
required. 

Example : A^Tiat width of face for gear 4 P ? Divid- 
ing 8 by 4 and adding ^'' we obtain 2J", for width of 
face. For change gears on lathes, w^here it is deska- 
ble not to have face very wide, the following rule can 
be used : 

Divide 4 by the diametral pitch and add \" . 

By the latter rule a 4 P change geaa- would have but 
11" face. 
Speed of Gear The Speed of gear cutters is subject to so many con- 
ditions that definite rules cannot be given. "We apj^end 
a table of average speeds. A coarse pitch cutter for 
pinion, 12 teeth, would usually be run slower than a» 
cutter for a large gear of same pitch. 



PROVIDENCE, R. I. 



5y 



TABLE OF AVERAGE SPEEDS FOR GEAR-CUTTERS. 



xi 

r 

2 




Turns per 

minute cutting 

Cast Iron. 


Turns per 
minute cutting 
Wrouglit Iron 

and Steel. 


Feed to One 

Turn of Cutter 

in Cast Iron. 


Feed to One 
Turn of Cutter 

in Wrought 
Iron and Steel. 


Feed per 
minute in 
Cast Iron. 


Feed per 

minute in 

Wrought Iron 

and Steel. 


5 in. 


24 


18 


.025 in. 


.011 in. 


.60 in. 


. 20 in. 


21 


41 u 


30 


24 


.028 " 


.013 " 


.84 " 


.31 " 


3 


m " 


36 


28 


.031 ^' 


.015 " 


1.12 " 


.42 " 


4 


3| " 


42 


32 


.084 " 


.017 " 


1.43 " 


.54 " 


6 


3tV " 


50 


40 


.037 " 


.019 " 


1.85 " 


.76 " 


6 


41" 


75 


55 


.030 " 


.016 " 


2.25 " 


.88 " 


7 


•4 " 


85 


65 


.032 " 


.018 " 


2.72 " 


1.17 ^' 


8 


'U " 


95 


75 


.034 " 


.020 " 


3.23 " 


1.50 " 


10 


2J " 


125 


90 


.026 " 


.014 " 


3.25 " 


1.26 " 


12 


2 " 


135 


]00 


.027 " 


.017 " 


3.64 " 


1.70 '* 


20 


1| " 


145 


115 


.029 " 


.021 " 


4.20 " 


2.41 " 


32 


1} " 


160 


135 


.031 '• 


.025 " 


4.96 " 


3.37 " 



In brass the speed of geai'-cutters can be twice as g^f^J/ ® ^ ^ ^ 
fast as in cast iron. Clock-makers and those making a 
specialty of brass gears exceed this rate even. A 12P 
cutter has been run 1,200 (twelve hundred) turns a 
minute in bronze. A 32 P cutter has been run 7,000 
(seven thousand) tiu'ns a minute in soft brass. 

In cutting 5 P cast-iron gears, 75 teeth, a No. 1, (^ ^ from F^L^tu^l^ 
cutter was run 136 (one hundred and thirty-six) turns 
a minute, roughing the spaces out the full 5 P depth ; 
the teeth w^ere then finished watli a 5 P cutter, running 
208 (two hundred and eight) turns a minute, feeding 
by hand. The cutter stood w^ell, but, of course, the 
cast iron was quite soft. A 4 P cutter has finished 
teeth at one cut, in cast-u'on gears, 86 teeth, running 48 
(forty-eight) turns a minute and feeding j\" at one 
turn, or 3 in. in a minute. 

Hence, while it is generally safe to run cutters as in 
the table, yet w^hen many gears are to be cut it is w^ell to 
see if cutters will stand a higher speed and more feed. 

In gears coarser than 4 P it is more economical to 
first cut the full depth with a stocking cutter and then 
finish with a gear cutter. This stocking cutter is made 



60 BEOWN & SHARPE MFG. CO. 

on tlie principle of a circular splitting saw for wood. 
The teeth, however, are not set ; but side relief is ob- 
tained by making sides of cutter blank hollowing. The 
shape of stocking cutter can be same as bottom of 
spaces in a 12-tooth gear, and the thickness of cutter 
can be J of the cu'cular pitch. 

Keep Cutters The matter of keeping cutters sharj) is so important 
that it has sometimes been found best to have the work- 
man grind them at stated times, and not wait until he 
can see that the cutters are dull. Thus, have him 
grind every two hours or after cutting a stated number 
of gears. Cutters of the style that can be ground 
upon their tooth faces without changing form are rap- 
idly destroyed if allowed to run after they are dull. 
Cutters are oftener wasted by trying to cut with them 
when they are dull than by too much grinding. Grind 
the faces radial with a free cutting wheel. Do not let 
the wheel become glazed, as this will di'aw the temper 
of the cutter. 

In Chapter VI. was given a series of cutters for cut- 
ting gears having 12 teeth and more. Thus, it was 
there implied that any gear of same pitch, having 135 
teeth, 136 teeth, and so on up to the largest gears, and, 
also, a rack, could be cut with one cutter. If this cut- 
ter is 4 P, we would cut with it all 4 P gears, having 
135 teeth or more, and we would also cut with it a 4 P 
rack. Now, instead of always referring to a cutter by 
the number of teeth in gears it is designed to cut, it 
has been found convenient to designate it by a letter 
or by a number. Thus, we call a cutter of 4 P, made 
to cut gears 135 teeth to a rack, inclusive, No. 1, 4 P. 
We have adopted numbers for designating involute 

Involute Gear Pfear-cutters a-3 in the f ollowincf table : 

•Cutters. 

No. 1 will cut wheels from 135 teeth to a rack inclusive. 



55 




134 teeth 


35 




54 " 


26 




34 " 


21 




25 " 


1? 




20 " 


14 




16 " 


U 




13 " 



PROVIDENCE, R. I. Gl 

By this i^lan it takes eight cutters to cut all gears 
liaviug twelve teeth aud over, of any one pitch. 

Thus it takes eight cutters to cut all involute 4 P 
gears having twelve teeth and more. It takes eight 
other cutters to cut all involute gears of 5 P, having 
12 teeth and more. A No. 8, 5 P cutter cuts only 5 P 
gears having 12 and 13 teeth. A No. G, 10 P cutter 
cuts only 10 P gears having 17, 18, 19 and 20 teeth. 
On each cutter is stamped the number of teeth at the 
limits of its range, as well as the number of the cutter. 
The number of the cutter relates only to the number 
of teeth in gears that the cutter is made for. 

In ordering cutters for involute spur-gears two things 
must be given : 

1. Either the number of teeth to be cut in the gear ^ iiow to order 

^ ^ Involute C u t - 

or the number of the cutter, as given in the foregoing ^^^^' 
table. 

2. Either the pitch of the gear or the diameter and 
number of teeth to be cut in the gear. 

If 25 teeth are to be cut in a 6 P involute gear, the 
cutter will be No. 5, G P, which cuts all G P gears from 
21 to 25 teeth inclusive. If it is desired to cut gears 
from 15 to 25 teeth, three cutters will be needed, No. 
5, No. 6 and No. 7 of the pitch required. If the pitch 
is 8 and gears 15 to 25 teeth are to be cut, the cutters 
should be No. 5, 8 P, No. G, 8 P, and No. 7, 8 P. 

For each pitch of epicycloidal, or double-curve gears, , Epicycioidai 
■24 cutters are made. In coarse-pitch gears, the varia- ci^rve cutters, 
tion in the sliajDe of spaces between gears of consecu- 
tive-numbered teeth is greater than in fine-pitch gears. 

Having a set of cutters for each pitch, to consist of 
so large a number as 24, has been established because 
double curve teeth have generally been preferred in 
coarse-pitch gears. 

Double-curve gear-cutters are designated by letters 
instead of by numbers ; this is to avoid confusion in 
ordering. 

Following is the list of epicycloidal or double-curve 
gear-cutters. 



62 



cycloid a 1 or 
Double -curve 
Gear Cutters. 







BROUT? & 


SHARPE 


MFG. 


CO. 










tte 


T A cuts 12 


teeth. 


Cutter M cuts 27 to 29 teei 


u 


B 




13 




a 


N 


a 


30 


u 


33 " 


u 


C 




14 




u 





u 


34 


li 


37 " 


u 


D 




15 




a 


P 


u 


38 


ii 


42 *• 


li 


E 




16 




u 


Q 


u 


43 


ii 


49 " 


*' 


F 




17 




u 


R 


(( 


50 


ii 


59 " 


u 


G 




18 




a 


S 


ii 


60 


ii 


74 *' 


a 


H 




19 




u 


T 


a 


75 


ii 


99 " 


(( 


I 




20 




(( 


U 


a 


100 


ii 


149 " 


u 


J 




21 to 22 


a 


V 


a 


150 


ii 


249 " 


n 


K 




23 to 24 


li 


w 


a 


250 


u 


Rack. 


a 


L 




24 to 26 


a 


X 


a 


Rack. 





A cutter that cuts more than one gear is made of 
proper form for the smallest gear in its range. Thus, 
cutter J for 21 to 22 teeth is correct for 21 teeth; 
cutter S for 60 to 74 teeth is correct for 60 teeth, 
and so on. 

Epicycioidai ^^ Ordering epicycloidal gear-cutters designate the 

Cutters. letter of the cutter as in the foregoing table, also 

either give the pitch or give data that will enable us 
to determine the pitch, the same as directed for invo- 
lute cutters. 

More care is requu'ed in making and adjusting epi- 
cycloidal gears than in making involute gears. 

B?vV? ^ear ^^ Ordering bevel gear cutters three things must be 

Cutters. given : 

1. 7%e numher of teeth in each gear. 

2. Either the pitch of gears or the largest pitch 

diameter of each gear ; see Fig- 16. 

3. The length of tooth face. 

If the shafts are not to run at right angles, it 
should be so stated, and the angle given. Involute 
cutters only are used for cutting bevel gears. No at- 
tempt should be made to cut epicycloidal tooth bevel 
gears. 
How to order j^ ordering worm-wheel cutters, three things must 

Worm -gear o ' ° 

Cutters. "be given : 

1. Niimher of teeth in the loheel. 

2. Pitch of the worm; see Chapter JCI. 

3. Whole diameter of worm. 



PROVIDENCE, R. I. 63 

In any order connected -with geai's or gear-cutters, 
T^'hen the word " Diameter" occurs, we usually under- 
stand that the pitch diameter is meant. When the 
ichole diameter of a gear is meant it should be plainly 
written. Cai'e in giving an order often saves the delay 
of asking fui'ther instructions. An order for one gear- 
cutter to cut from 25 to 30 teeth cannot be filled, be- 
cause it takes two cutters of involute form to cut from 
25 to 30 teeth, and three cutters of epicycloidal form 
to cut from 25 to 30 teeth. 

Sheet zinc is convenient to sketch gears upon, and 
also for making templets. Before making sketch, it is 
well to give the zinc a dark coating with the following 
mixture: Dissolve 1 ounce of sulphate of copper (blue 
vitriol) in about 4 ounces of water, and add about one- 
half teaspoonful of nitric acid. Apply a thin coating 
with a piece of waste. 

This mixture will give a thin coating of copper to 
iron or steel, but the work should then be rubbed dry. 
Care should ba taken not to leave the mixture where it 
is not wanted, as it rusts iron and steel. 



64 



BROWN & SUARPE MFG. CO. 



GEAR WHEELS. 

TABLE OF TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN. 



p 

2 


Threads or 

Teeth per inch 

Linear. 


1- 
q 


Thickness of 

Tooth on 
Pitcli Line. 


a 


•B 
p- . 

bfjO 

P 


Depth of Space 

below 

Pitch Line. 


1^ 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


J '' 
i 


p 


t 


s 


D" 


s+f 


D"+/ 
1.3732 


P'x.3lP'x.3:5 


1.5708 


1.0000 


.6366 


1.2732 


.7366 


.6200.6700 


li 


A 


1.6755 


.9375 


.5968 


1.1937 


.6906 


1.2874 


.5813 .6281 : 


If 


i 


1.7952 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5425! 5863 1 


If 


A 


1.9333 


.8125 


.5173 


1.0345 


.5985 


1.1158 


.5038 .5444 i 


li 


1 


2.0944 


.7500 


.4775 


.9549 


.5525 


1.0299 


.4650 .5025 \ 


IxV 


M 


2.1855 


.7187 


.4576 


.9151 


.5294 


.9870 


.4456 


.4816 1 


If 


t\ 


2.2848 


.6875 


.4377 


.8754 


.5064 


.9441 


.4262 


.4606 ' 


l.v 


iS- 


2.3936 


.6562 


.4178 


.8356 


.4834 


.9012 


.4069 .4397 


li 


i 


2.5133 


.6250 


.3979 


.7958 


.4604 


.8583 


.3875 .4188 


lA 


H 


2.6456 


.5937 


.3780 


.7560 


.4374 


.8156 


.3681 .3978 


li 


1 


2.7925 


.5625 


.3581 


.7162 


.4143 


.7724 


.3488 


3769 


^-h 


if 


2.9568 


.5312 


.3382 


.6764 


.3913 


.7295 


.3294 .3559 


1 


1 


3.1416 


.5000 


.3183 


.6366 


.3683 


.6866 


.3100 .3350 


if 


iiV 


3.3510 


.4087 


.2984 


.5968 


.3453 


.6437 


.2906 .3141 


^ 


u 


3.5904 


.4375 


.2785 


.5570 


.3223 


.6007 


.2713 .2931 


ii 


ifV 


3.8666 


.4062 


.2586 


.5173 


.2993 


.5579 


.2519 .2722 


a 

4 


li 


4.1888 


.3750 


.2387 


.4775 


.2762 


.5150 


.2325 .2513 


H 


ifV 


4.5696 


.3437 


.2189 


.4377 


.2532 


.4720 


.2131 .2303 


1 


li 


4.7124 


.3333 


.2122 


.4244 


.2455 


.4577 


.2066 .2233 



PROVIDENCE, R. I. 



65^ 



TABLE OF TOOTH VAItTS.—Co7iti?iued. 

CIRCULAR PITCH IN FIRST COLUMN. 



Is 


Threads or 

Teeth per inch 

Linear. 


II 

5^ 


Tiiickness of 
«• Tooth on 
Pitch Line. 


< 


II 


Deplh of Space 

below 

Pitch Line. 




Widtli of 

Thread -Tool 

at End. 


Width of 
Thread at Top. 


p' 


f 


P 


.<? 


D' 


^^f 


D"+/. 


Px.31 


P'x.335 


s 


n 


5.0265 


.3125 .1989 


.3979 


.2301 


.4291 


.1938 


.2094 


A 


n 


5.5851 


.2812 


.1790 


.3581 


.2071 


.3862 


.1744 


.1884 


i 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 


.3433 


.1550 


.1675 


tV 


2f 


7.1808 


.2187 


.1393 


.2785 


.1611 


.3003 


.1356 


.1466 


1 


2i 


7.8540 


.2000 


.1273 


.2546 


.1473 


.2746 


.1240 


.1340 


§ 


2| 


8.3776 


.1875 


.1194 


.2387 


.1381 


.2575 


.1163 


.1256 


i 


3 


9.4248 


.1666 


.1061 


.2122 


.1228 


.2289 


.1033 


.1117 


,v 


3i 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0969 


.1047 


f 


3i 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0886 


.0957 


i 


4 


12.5664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0775 


.0838 


1 


4i 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0689 


.0744 


i 


5 


15.7080 


.1000 


.0637 


.1273 


.0737 


.1373 


.0620 


.0670 


3 


5i 


16.7552 


.0937 


.0597 


.1194 


.0690 


.1287 


.0581 


.0628 


i 


6 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0517 


.0558 


1 


7 


21.9911 


.0714 


.0455 


0910 


.0526 


.0981 


.0443 


.0479 


i 


8 


25.1327 


.0625 


.0398 


.0796 


.0460 


.0858 


.0388 


.0419 


i 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0372 


tV 


10 


31.4159 


.0500 


.0318 .0637 


.0368 


.0087 


.0310 


.0335 


iV 


IG 


50.2655 


.0312 .0199'. 0398 


.0230 


.0429 


.0194 


.0209 



66 



BROWN & SHARPE MFG. CO. 



GEAK WHEELS. 

TABLE OF TOOTH PARTS DIAMETRAL PITCH IN FIRST COLUMN. 



1 . 

II 

5^ 




Thickness 
of Tooth on 
Pitch Line. 




rv 

r 


Depth of Space 

below 

Pitch Line. 


OH 


P 


P' 


t 


s 


D" 


s+f. 


D"+/. 


4 


6.2832 


3.1416 


2.0000 


4.0000 


2.3142 


4.3142 


f 


4.1888 


2.0944 


1.3333 


2.6666 


1.5428 


2.8761 


1 


3.1416 


1.5708 


1.0000 


2.0000 


1.1571 


2.1571 


li 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


li 


2.0944 


1.0472 


.6666 


1.3333 


.7714 


1.4381 


If 


1.7952 


.8976 


.5714 


1.1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2i 


1.3963 


.6981 


.4444 


.8888 


.5143 


.9587 


2i 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


2^ 


1.1424 


.5712 


.3636 


.7273 


.4208 


.7844 


3 


1.0472 


.5236 


.3333 


.6666 


.3857 


.7190 


^ 


.8976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


.7854 


.3927 


.2500 


.5000 


.2893 


.5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


1 6 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


7 


.4488 


.2244 


.1429 


.2857 


.1653 


.3081 


8 


.3927 


.1963 


.1250 


.2500 


.1446 


.2696 


9 


.3491 


.1745 


.1111 


.2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 

1 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


.0833 


.1666 


.0964 


.1798 


13 


.2417 


.1208 


.0769 


.1538 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


.1429 


.0826 


.1541 



PROVIDENCE, R. I. 



67 



TABLE OF TOOTH VARTS—Cofitmued. 

DIAMETRAL PITCH IN FIRST COLUMN. 



Diametral 
Pitch. 




Thickness 
of Tooth on 
Pitch Line. 


Addendum 
and ^ 


Working Depth 
of Tooth. 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


P. 


P'. 


t. 


s. 


D". 


s+f. 


.1438 


15 


.2094 


.1047 


.0666 


.1333 


.0771 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.05-8 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.0643 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


^0 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


21 


.1309 


.0654 


.0417 


.0833 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0413 


.0770 


30 


,1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


31 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


,0393 


.0250 


.0500 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


,0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0333 


.0193 


.0360 



PART IL 



• CHAPTER I. 
TANGENT OF ARC AND ANGLE. 



In Part II. we sliall sliow how to calculate some^J;;;^f^«f^t«^« 
of the functions of a right:angle triangle from a table ^ 
of cii'cular functions, the application of these calcula- 
tions in some chapters of Part I. and in sizing blanks 
and cutting teeth of spiral gears, the selection of 
cutters for spiral gears, the application of continued 
fractions to some problems in gear wheels and cutting 
odd screw-threads, etc., etc. 

A Function is a quantity that depends upon another 
quantity for its value. Thus the amount a workman 
earns is a function of the time he has worked and of ^J^^^^ 
his wages per hour. 



Function de- 




In any right-angle triangle, O A B, we shall, f o^ ^H^^^t- angle 
convenience, call the two lines that form the right 
angle O A B the sides, instead of base and perpen- 
dicular. Thus O A B, being the right angle we call 
the line O A a side, and the line A B a side also. 

When we speak of the angle A O B, we call the line 
O A the side adjacent. AMien we are speaking of the '^^'^^ adjacent, 
angle A B O we call the line AB the side adjacent. 
The line opposite the right angle is the hypothenuse. Hypothenuse. 



70 

Tangent. 



BROWN & SHARPE MFG. CO. 

The Tangent of an arc is the line that touches it at 
one extremity and is terminated by a line drawn from 
the center through the other extremity. The tangent 
is always outside the arc and is also perpendicular to 
the radius which meets it at the point of tangency. 




Fig. 30 

Thus, in Fig. 30, the line A B is the tangent of the arc 
A C. The point of tangency is at A. 

An angle at the center of a circle is measured by the 
arc intercepted by the sides of the angle. Hence the 
tangent A B of the arc A C is also the tangent of the 
angle A O B. 

In the tables of circular functions the radius of the 
arc is unity, or, in common practice, we take it as one 
inch. The radius O A being 1", if we know the length 
of the line or tangent A B w^e can, by looking in a 
table of tangents, find the number of degrees in the 
angle A O B. 
To find the Thus, if A B is 2.25" long, we find the angle A O B 

Degrees in an *=* '^ 

Angle. is 66 very nearly. That is, having found that 2.2460 

is the nearest number to 2.25 in the table of tangents 
at the end of this volume, we find the corresponding 
degrees of the angle in the column at the left hand of 
the table and the minutes to be added at the top of 
the column containing the 2.2460. 

The table gives angles for every 10', which is suf- 
ficient for most purposes. 



PROVIDENCE, R. I. 71 

Now, if we Lavo a right-angle triangle with an angle 
the same as O A B, but with O A two inches long, the 
line A B will also be twice as long as the tangent of 
angle A O B, as found in a table of tangents. 

Let us take a triangle with the side O A = 5" long, fina^^^iTo^^ d «! 
and the side A B = 8" long ; what is the number of |;;^f^« ^^ ^"^ 
degrees in the angle A O B ? 

Dividing 8" by 5 we find what would be the length 
of A B if O A was only 1" long. The quotient then 
would be the length of tangent when the radius is 1" 
long, as in the table of tangents. 8 divided by 5 is 
l.G. The nearest tangent in the table is 1.G003 and 
the corres2)onding angle is 58°, which would be the 
angle A O B when A B is 8" and the radius O A is 5" 
very nearl3\ The diiference in the angles for tangents 
l.C)003 and 1.6 could hardly bs seen in practice. The 
side opposite the required acute angle corresponds to 
the tangent and the side adjacent corresponds to the 
radius. Hence the rule : 

To find the tangent of either acute angle in a right- rpj^g^^^*^ *^® 
angle triangle : Divide the side (pposlte the angle by 
the side adjacent the angle and the quotient lolll le 
the tangent of the angle. This rule should be com- 
mitted to memor}'. Having found the tangent of the 
angle, the angle can be taken from the table of tan- 
gents. 

The complement of an angle is the remainder after Complement 
subtracting the angle from 90°. Thus 40° is the com- 
jDlement of 50°. 

The Cotangent of an angle is the tangent of the cotangent, 
complement of the angle. Thus, in Fig. 30, the line 
A B is the cotangent of A O E. In right-angle tri- 
angles either acute angle is the complement of the 
other acute angle. Hence, if we know one acute angle, 
by subtracting this angle from 90° we get the other 
acute angle. As the arc approaches 90° the tangent 
becomes longer, and at 90° it is infinitely long. 

The sign of infinity is go. Tangent 90° = a>. 



72 BEOWN & SHAEPE MPCi. CO. 



CHAPTER II. 

APPLICATION OF. TABLE OF TANGEHTS TO PROBLEMS OCCURRING IN 
MACHINE CONSTRUCTION— ANGLES OF BEVEL GEARS. 



To avoid confusion we will illustrate one gear only. 
The same rules apply to all sizes of bevel gears. Fig. 
31 is the out.ine of a pinion 4 P, 20 teeth, to mesh with 
•a gear 28 teeth, shafts at right angles. For making 
sketch of bevel gears see Chapter IX., Part I. 

In Fig. 31, the line O in^ m is continued to the line 
a h. The angle c' O i that the cone pitch-line makes 
with the center line may be called the center angle. 
Angle of The center angle c O i is equal to the angle of edge 
' c' i c. G i is the side opposit3 the center angle c' O 
i, and c' O is the side adjacent the center angle, c' 
i = 2.5"; c' O = 3.5". Dividing 2.5" by 3.5" we 
obtain .71428" + as the tangent of c' O ^. In the table 
we find .71329 to be the nearest tangent, the corre- 
sponding angle being 35° 30'. 35 J°, then, is the center 
angle c O i and the angle of edge c i n, very nearly. 
When the axes of bevel gears are at right angles the 
' angle of edge of one gear is the comj^lement of angle 
of edge of the other gear. Subtracting, then, 35J° 
from 90° we obtain 54J° as the angle of edge of gear 
28 teeth, to mesh with gear 20 teeth, Fig. 31. 
Angle of Pace. To obtain angle of face O m" c\ the distance c O 
becomes the side opposite and the distance m" c is 
the side adjacent. 

The distance c O is 3.5", the radius of the 28 tooth 
bevel gear. The distance c m" is by measurement 
.2.82". 

Dividing 3.5 by 2.82 we obtain 1.2411 for tangent 
of angle of face O m" c . The nearest tangent in the 
table is 1.2422 and the corresponding angle is 51° 10'. 



PROVIDENCE, R. I. 



73 



-^.1^ 




I ig. 31 



74 BROWN & SHARPE MFG. CO. 

To obtain cutting angle c O ii' we divide the distance 
c n' by c O. By measurement c' n" is 2.2". Divid- 
ing 2.2 by 3.5 we obtain .62857 for tangent of cutting 
angle. The nearest corresponding angle in the table 
is 32° 10'. 
Angie^^by"\he "^J ^ table of tangents, angles can be laid out upon 
Tangent _ E^x - gjieet zinc, ctc. This is often an advantage, as it is not 
convenient to lay protractor flat down so as to mark 
angles up to a sharp point. If we could lay off the 
length of a line exactly we could take tangents direct 
from table and obtain angle at once. It, however, is 
generally better to multiply the tangent by 5 or 10 
and make an enlarged triangle. If, then, there is a 
slight error in laying off length of lines it will not 
make so much difference with the angle. 

Let it be required to lay off an angle of 14° 30'. By 
the table we find the tangent to be .25861. Multiply- 
ing .25861 by 5 we obtain, in the enlarged triangle, 
1.29305" as the length of side opposite the angle 14° 
30'. As we have made the side o^^posite five times as 
large, we must make the side adjacent five times as 
large, in order to keep angle the same. Hence, Fig. 
32, draw the line A B 5" long ; perpendicular to this 
line at A draw the line A O 1.293" long ; now di'aw the 
line O B, and the angle ABO will be 14° 30'. 

If special accuracy is required, the tangent can be 
multiplied by 10; the line A O will then be 2.586'' long 
and the line A B 10" long. Remembering that tbe 
acute angles of a right-angle triangle are the comple- 
ments of each other, Ave subtract 14° 30' from 90' and 
obtain 75° 30' as the angle of A O B. 

The reader will remember these angles as occurring 
in Part I., Chapter IV., and obtained in a different 
way. A semicircle upon the line B touching the 
extremities O and B will just touch the right angle at 
A, and the line O B is four times as long as O A. 

Let it be requked to turn a piece 4" long, 1" diam- 
eter at small end, with a taper of 10° one side with the 
other ; what will be the diameter of the piece at the 
large end ? 

A section. Fig. 33, through the axis of this piece is 



PROVIDENCE, R. I. 



75 




/V ^0' 

-h6993^^- H 



F-ig. 32. 



Fis. 33 



76 BKOWN & SHARPE MFG. CO. 

To calculate ^]^q same as if we added two rio-ht-anofle trianj^les, O 

Diameter of a o o o ' 

T.a peri n gA B and O' A' B', to a stiwht pieca A' A B B', 1" 

piece. Fig. 33. _ ' o l > 

wide and 4 long, the acute angles B and B' being 5"^, 
thus making the sides O B and O' B' 10° with each 
other. 

The tangent of 5° is .08748, which, multiplied by 
4", gives .34992" as the length of each line, A O and 
A' O', to be added to 1" at the large end. Taking 
twice .34992" and adding to 1" we obtain 1.69984" as 
the diameter of large end. 

These chapters, I. and II., must be thoroughly 
studied before taking up the next chapters. If once 
the memory becomes confused as to the tangent and 
sine of an angle, it will take much longer to get righted 
than it will to first carefully learn to recognize the 
tangent of an angle at otice. 

If one knows what the tangent is, he can better tell 
the functions that are 7iot tangents. 



PROVIDENCE, R. I. 



77 



CHAPTER III. 

SINE— COSINE AND SECANT : SOME OF THEIR APPLICATIONS IN 
MACHINE CONSTRUCTION. 



llie Sine of an arc is tlie line drawn from one 
extremity of the arc to the diameter passing through 
the other extremity, the line being perpendicular to 
the diameter. 

Another definition is : The sine of an arc is the dis- 
tance of one extremity of the arc from the diameter, 
through the other extremity. 

The sine of an angle is the sine of the arc that 
measures the angle. 

In Fiff. 34, A C is the sine of the 



Sine of Arc 
and Angle. 



B C, and of 

the angle B O C. It will be seen that the sine is 
Always inside of the arc, and can never be longer than 
the radius. As the arc ap- 
proaches 90°, the sine comes 
nearer to the radius, and at 90^ 
the sine is equal to 1, or is the 
radius itself. From the defini- 
tion of a sine, the side A C, 
opposite the angle A O C, in 
any right-angle triangle, is the 
sine of the angle A O C, when 
O C is the 7'adius of the arc. 
Hence the rule : In any right-angle triangle, the side To find the 
opposite either acute angle, divided by the hypothe- 
mcse, is equal to the sine of the angle. 

The quotient thus obtained is the length of side 
opposite the angle when the hypothenuse or radius is 
unity. The rule should be carefully committed to 
memory. 




Fig. 3-i 



Sine. 



78 



BROWN & SHARPE MFG. CO. 



Arc^^^^ ^^ ^^ ""^ 67iorc? is a straight line joining the extremities of 
an arc, and is twice as Jong as the sine of half the 
angle measured by the arc. Thus, in Fig. 34, the 
chord F C is twice as long as the sine A C. 




Fig. 35 

Let there be four holes equidistant about a circle 
3" in diameter — Fig. 35 ; what is the shortest distance 
between two holes ? This shortest distance is the 

flnd^thTchord! chord A B, wdiich is twice the sine of the angle COB. 
The angle A O B is one -quarter of the circle, and 
C O B is one-eighth of the circle. 360°, divided by 
8=45°, the angle COB. The sine of 45° is .70710, 
which, multiplied by 1.5", gives length C B in the 
cu'cle, 3" in diameter, as 1.06065". Twice this length. 
is the required distance A B=2.1213" 

When a cylindi'ical piece is to be cut into any num- 
ber of sides, the foregoing operation can be apj^lied to 
obtain the width of one side. A plane figure bounded 

Polygon. by straight lines is called a polygon. 



PROVIDENCE, R. I. 



79 



When the outside diameter and the number of sides of 

a regular polygon are given, to find the length of 

one of the sides : Divide ^(^0° by twice the number of ^o find the 

^ ^ loiigth of Side. 

sides ; multiply the si?ie of the quotient by the outer 

diameter, and the product loill be the length of 07ie of 

the sides. 

Multiplying by the diameter is the same as multi- 
plying by the radius, and that product again by 2. 

!/7ie Cosine of an angle is the sine of the comple- cosine. 
ment of the angle. 

In Fig. 34, C O D is the complement of the angle 
A O C ; the line C E is the sine of COD, and hence 
is the cosine of B O C. The line O A is equal to C E. 
It is quite as well to remember the cosine as the part 
of the radius, from the center that is cut off by the 
sine. Thus the sine A C of the angle A O C cuts off 
the cosine O A. The line O A may be called the 
cosine because it is equal to the cosine C E. 

In any right-angle triangle, the side adjacent either 
:acute angle corresponds to the cosine when the 
hypothenuse is the radius of the arc that measures 
the angle; hence: Divide the side adjacent the acute To and the 
angle by the hypothenuse^ and the quotient vxill be the 
■cosine of the angle. 

When a cylindrical piece is cut into a j^olygon of 
any number of sides, a table of cosines can be used to .Length of 

'^ _ ' _ sides of Poly- 

get the diameter across the sides. son. 




Fig. 30 



80 



BROWN 



SHARPS MFG. CO. 



Let a cyliuder, 2" diameter, Fig. 36, be cut six-sided ; 
what is the diameter across the sides ? 

The angle A O B, at the center, occupied by one of 
these sides, is one-sixth of the circle, =60°. The 
cosine of one-half this angle, 30°, is the line C O; 
twice this line is the diameter across the sides. The 
cosine of 30° is .86602, which, multii^lied by 2, gives 
1.73204" as the diameter across the sides. 

Of course, if the radius is other than unity, the cosine 
should be multiplied by the radius, and the product 
again by 2, in order to get diameter across the sides ; 
or what is the same thing, multiply the cosine by the 
whole diameter or the diameter across the corners. 

The rule for obtaining the diameter across sides of 
sides of a Poly- regular polygon, when the diameter across corners is 
given, will then be : Multiply the cosine of 360° 
divided hy tioice the nmnher of sides, by the diameter 
across corners, and the product will he the diameter 
across sides. 

Look at the right-hand column for degrees of the 
cosine, and at bottom of page for minutes to add to- 
the degrees. 

2'he Secant of an arc is a straight line drawn from 
the center through one end of an arc, and terminated 
by a tangent drawn from the other end of the arc. 

Thus, in Fig. 37, the line O B is the secant of the 
angle COB. 

A C B 



Kule for Di- 
ameter across 



Secant. 




Fig. 37 

To find the In any right-angle triangle, divide the hypothenuse 
hy the side adjacent either acute angle, and the quo- 
tient loill he the secant of that angle. 



PROVIDENCE, R. I. 81 

That is, if we divide the distance O B by O C, in 
the right-angle triangle COB, the quotient will be 
the secant of the angle COB. 

The secant cannot bo less than the radius ; it in- 
creases as the angle increases, and at 90° the secant is 
infinity =00 . 

A six-sided piece is to be l^r" across the sides ; how t^ 7^ T>^"n\ ^^ ^ 
large must a blank be turned before cutting the sides ? Jf ^a?oi^ gon ^^' 
Dividing 360° by twice the number of sides, we have 
30°, which is the angle COB. The secant of 30° is 
1.1547. 

The radius of the six-sided piece is .75". 

Multiplying the secant 1.1547 by .75", we obtain the 
length of radius of the blank O B ; multiplying again 
by 2, we obtain the diameter 1.732" 4-. 

Hence, in a regular polygon, when the diameter 
across sides and the number of sides are given, to find 
diameter across corners : Multiply the secant of 8G0° 
divided hy ticice the number of sides, hy the diameter 
across sides, and the product vnll he the diameter 
across corners. 

It will be seen that the side taken as a divisor has 
been in each case the side corresponding to the radius 
of the arc that subtends the angle. 



82 BEOWS k SHARPE MFO. CO. 



CHAPTER IV. 

PARTHER APPLICATION OF CIRCULAR FUKCTIONS— WHOLE 

DIAMETER OF BEVEL GEAR BLABKS— FACE ANGLE 

OF BEVEL GEAR BLANKS. 



The largest pitch diameter, kj, of a bevel gear, as in 
Fig. 38, is known the same as the pitch diameter of 
any spur gear. Now, if we know the distance b o or 
its equal a q, we can obtain the whole diameter of 
bevel gear blank by adding twice the distance b o to 
the largest pitch diameter. 
crem^?^Vig' Twice the distance b o, or what is the same thing, 
^^- the sum oi a q and ^ o is called the diameter incre- 

ment, because it is the amount by which we increase 
the largest j)itch diameter to obtain the whole or out- 
side diameter of bevel gear blanks. The distance b o 
can be calculated without measuring the diagram. 

The angle b o j i^ equal to the angle of edge. 

The angle of edge, it will be remembered, is the 
angle formed by outer edge of blank or ends of teeth 
with the end of hub or a plane perpendicular to the 
axis of gear. 

The distance ^ o is equal to the cosine of angle of 
edge, multiplied by the distance j o. The distance j o 
is the addendum, as in previous chapters ( = s). 

Hence the rule for obtaining the diameter increment 
of any bevel gear: Multiply the cosine of angle of 
edge by the toorking depth of teeth (D"), aiid the 
product will be the diameter increment. 

By the method given in Chapter II. we find the 

angle of edge of gear (Fig. 38) is 56° 20'. The cosine 

of 56° 20° is .55436, which, multiplied by |", or the 

^Outside Diam- depth of the 3 P gear, gives the diameter increment of 

the bevel gear 18 teeth, 3 P meshing with pinion of 12 



PKO^^DENCE, R. I. 



83 




84 BKOWN & SHARPE MFG. CO. 

teeth, f of .55436=. 369" + (or .37", nearly). Adding 
the diameter increment, .37", to the largest pitch 
diameter of gear, 6", we have 6.37" as the outside 
diameter. 

In the same manner, the distance c d is half the 
diameter increment of the pinion. The angle c d h is 
equal to the center angle of pinion, and when axes are 
at right angles is the complement of center angle of 
gear. The center angle of pinion is 33° 40'. The 
cosine, multiplied by the working depth, gives .555" 
for diameter increment of j)iiiion, and we have 4.555" 
for outside diameter of pinion. 

In tui'ning bevel gear blanks, it is sufficiently accu- 
rate to make the diameter to the nearest hundredth of 
an inch. 
Angle incre- The Small angle o O J is called the angle increment. 
"When shafts are at right angles the face angle of one 
gear is equal to the center angle of the other gear, 
tninus the angle increment. 

Thus the angle of face of gear (Fig. 38) is less than 
the center angle D O k, or its equal O j k by the angle 
o OJ. That is, subtracting o Oj from OJ k, the re- 
mainder will be the angle of face of gear. 

Subtracting the angle increment from the center 
angle of gear, the remainder will be the cutting 
angle. 

The angle increment can be obtained by dividing 
oJ, the side opposite, by Oj, the side adjacent, thus 
finding the tangent as usual. 

The length of cone-pitch line from the common 
center, O to j, can be found, without measuring dia- 
gram, by multiplying the secant of angle Oj k, or the 
center angle of pinion, by the radius of largest pitch 
diameter of gear. 

The secant of angle 0^ k, 33° 40', is 1.2015, which, 
multiplied by 3", the radius of gear, gives 3.6045" as 
the length of line O j. 

Dividing oj by Oj, we have for tangent .0924, and 
for angle increment 5° 20'. 

The angle increment can also be obtained by the 
following: rule : 



PROVIDENCE, R. I. 



85 



Divide the sine of center angle by half the num- 
ber of teeth, and the quotient will be the tangent of 
increment angle. 

Subtracting the angle increment from center angles 
of gear and pinion, we have respectively : 
Cutting angle of gear, 51°. 
Cutting angle of pinion, 28° 20'. 

Bemembering that when the shafts are at right 
angles, the face angle of a gear is equal to the cutting 
angle of its mate (Chapter X.), we have: 
Face angle of gear, 28° 20'. 
Face angle of pinion, 51°. 

It will be seen that both the whole diameter and the 
angles of bevel gears can be obtained without making 
a diagram. Mr. George B. Grant has made a table of 
different pau'S of gears from 1 to 1 up to 10 to 1, con- 
taining diameter increments, angle increments and 
center angles, and has published it in the American 
Machinist of October 31, 1885. We have adopted the 
terms " diameter increment," " angle increment " and 
" center angle " from him. He uses the term " back 
angle^^ for what we have called angle of edge. 

In laying out angles, the following method may be to lay out au 
preferred, as it does away with the necessity of making sine.*^ ^ 
a right angle : 

Draw a circle, ABO (Fig. 39), ten inches in diam- 
eter. Set the dividers to ten times the sine of the 
required angle, and point off this distance in the cir- 
cumference as at A B. From any point O in the cir- 




:Fi.g.39 



SG BROWN & SHARPE MFG. CO. 

cumference, di'aw tlie lines O A and O B. The angle 
A O B is the angle required. Thus, let the required 
angle be 12°. The sine of 12° is .20791, which, multi- 
l)hed by 10, gives 2.0791", or 2^" nearly, for the 
distance A B. 

Any diameter of circle can be taken if we multiply 
the sine by the diameter, but 10" is very convenient, 
as all we have to do with the sine is to move the 
decimal point one j)lace to the right. 

If either of the lines, as O B, passes through the 
center, the angle O A B will be a right angle. 



PROVIDENCE, R. I. 87 



CHAPTER V. 
SPIRAL GEARS— CALCDLATIOHS FOR PITCH OF SPIRALS. 



When the teeth of a gear are cut, not in a straight Spiral Gear. 
path, like a spur gear, but in a helical or screw- like 
path, the gear is called, technicall}", a twisted or screw 
gear, but more generally among mechanics, a spiral 
gear. A distinction is sometimes made between a 
screw gear and a twisted gear. In twisted gears the 
pitch surfaces roll upon each other, exactly like spur 
gears, the axes being parallel, the same as in Fig. 1, 
Part I. In screw gears there is an end movement, 
or slipping of the pitch surfaces upon each other, the 
axes not being parallel. In screw gearing the action 
is analogous to a screw and nut, one gear driving 
another by the end movement of its tooth path. This 
is readily seen in the case of a worm and worm-wheel, 
when the axes are at right angles, as the movement of 
wheel is then wholly due to the end movement of 
worm thread. But, as wg make the axes of gears more 
nearly parallel, they may still be screw gears, but the. 
distinction is not so readily seen. 

We can have two gears that are alike run together, 
with their axes at right angles, as at A B, Fig. 41. 

The same gear may be used in a train of screw gears 
or in a train of twisted gears. Thus, B, as it relates to 
A, may be called a screw gear ; but in connection with 
C, the same gear, B, may be called a twisted gear. 
These distinctions are not usually made, and we call 
all helical or screw-like gears made on the Universal 
Milling Machine spiral gears. 

When two external spiral gears run together, with Direction of 
their axes parallel, the teeth of the geai's must have erouce to Axes. 
opposite hand spirals. '^' 



OO BROWN & SHARPE MFG. CO. 

Thus, in Fig. 41 the gear B has right hand spiral 
teeth, and the gear C has left hand spiral teeth. When 
the axes of two spiral gears are at right angles, both 
gears must have the same hand spu-al teeth. A and 
B, Fig. 41, have right hand spiral teeth. If both gears 
A and B had left hand spiral teeth, the relative du'ec- 
tion in which they turn w^ould be reversed. 

Spiral Pitch. rj^j^^ spiral pitch or pitch of spiral is the distance the 

spiral advances in one turn. Strictly, this is the lead 
of the spiral. A cyhnder or gear cut with spiral 
grooves is merely a screw of coarse pitch or long lead; 
that is, a spiral is a coarse pitch screw, and a screw is 
a fine pitch sj^iral. 

Since the introduction and extensive use of the 
Universal Milling Machine, it has become customary 
to call any screw cut in the milling machine a spiral. 
The spu'al pitch is given as so many inches to one 
turn. Thus, a cylinder having a spiral groove that ad- 
vances six inches to one turn, is said to have a six inch 
spiral. 

In screws the pitch is often given as so many threads 
to one inch. Thus, a screw of -g" lead is said to be 
2 threads to the inch. The reciprocal expression is 
not much used with spirals. For example, it would 
not be convenient to speak of a spiral of 6" lead, as \ 
threads to one inch. 

The calculations for spirals are made from the func- 
tions of a right angle triangle. 
Example, Cut from paper a right angle triangle, one side about 

tuiTof^ He^fxthe right angle 6" long, and the other side about the 
pua . I'iglit angle 2". Make a cylinder 6" in circumference. 
It will be remembered (Part I., Chapter II.) that the 
circumference of a cylinder, multiplied by .3183, equals 
the diameter— 6" X. 3183 = 1.9098". Wrap the paiDer 
triangle around the cylinder, letting the 2" side be 
parallel to the axis, the 6" side perpendicular to the 
axis and reaching around the cylinder. The hypoth- 
eneuse now forms a helix or screw-like line, called 
a spiral. Fasten the paper triangle thus wrapped 
around. See Fig. 42. 



PEOVIDENCE, n. I. 



89 




Fig, 40.-RACKS AND GEARS. 




Fig. 41.-SPIRAL GEARING, 



90 



BROWN & SHARPE MFG. CO. 




:Fig. -t3 



If we now turn this cylinder A B C D in the direc- 
tion of the arrow, the spiral will advance from O to E. 
This advance is the pitch of the spiral. 

The angle EOF, which the spiral makes with the 
axis E O, is the angle of the spiral. This angle is 
found as in Chapter I. The circumference of the 
cylinder corresponds to the side opposite the angle. 
The pitch of the spiral corresponds to the side adjacent 
the angle. Hence the rule for getting angle of spiral : 
Rules for cai- Divide the circumference of the cylinder or spiral 

culating the j ^ o ± 

parts of a Spi- bg the number of inches of spiral to one turn., and the 
quotient icill he the tangent of angle of spiral. 

When the angle of spiral and circumference are 
given, to find the pitch : 

Divide the circumference by the tangent of angle, 
and the quotient will be the pitch of the spiral. 

When the angle of spiral and the lead or pitch of 
spiral are given, to find the cii'cumference : 

Multiply the tangent of angle by the pilch, and the 
product 10 ill be the circumference. 

Multiplying the circumference by .3183, we obtain 
the diameter of blank, as in Part I., Chapter II. 



PROVIDENCE, K. I. 91- 



CHAPTER VI. 

EXAMPLES IN CALCOLATION OF PITCH OF SPIRAL-ANGLE OF 

SPIRAL— CIRCUMFERENCE OF SPIRAL GEARS— 

A FEW HINTS ON CUTTING. 



It will be seen that the rules for calculating- circum- 
ference of spiral gears, angle and pitch of spii'al are 
the same as in Cliapters I. and II. for tangent and 
angle of a right angle triangle. In Chapter V. the 
word " circumference " is substituted for " side oppo-.* 
site," and the words " pitch of spiral " are substituted 
for "side adjacent." 

When two spiral sfears are in mesh the ano-le of Aggies of spi- 

ir o o rals with reter- 

spiral should be the same in one gear as in the other, ^^'^^ to Angle 

in order to have the shafts parallel and the teeth work 

properly together. When two gears both have right 

hand spiral teeth, or both have left hand spiral teeth, 

the angle of their shafts will be equal to the siwi of 

the angles of their spirals. But when two gears have 

different hand spirals the angle of their shafts will be 

equal to the difference of their angles of spirals. 

Thus, in Fig. 41 the gears A and B both have right 

hand spirals. The angle of both si)irals is 45°, their 

sum is 90°, or their axes are at right angles. But C 

has a left hand spiral of 45°. Hence, as the difference 

between angles of spirals of B and C is 0, their axes 

are parallel. 

When the two gears have the same number of teeth 
the pitch of the spiral will be alike in both gears. But 
when one gear has more teeth than the other the pitch 
of spiral in the larger gear should be longer in the 
same ratio. Thus, if one gear has 50 teeth and the pitch in spU 
other gear has 25 teeth, the pitch of spiral in the 5()i?iame'ter3!^^" 
tooth wheel should be twico as long as that of the 25 



92 BROWN & SHAEPE MFG. CO. 

tooth wheel. Of course, the diameter of pitch circle 
should be twice as large in the 50 tooth as in the 25 
tooth wheel. 

In spirals where the angle is 45° the circumference 

is the same as the spiral 2^itch, because the tangent of 

45° is 1. 

Circu^mfe?ence Sometimes the circumference is varied to suit a 

tosuitaSi>irai. pit(3li ^j^at can be cut on the machine and retain the 

angle required. This would apply to cutting rolls for 

making diamond-shaped impressions where the diam- 

•eter of the roll is not a matter of importance. 

When two gears are to run together in a given 
velocity ratio, it is well to first select spirals that the 
machine will cut of the same ratio, and calculate the 
numbers of teeth and angle to correspond. This will 
often save considerable time in figuring. 

The calculations for spiral gears present no special 
difficulties, but sometimes a little ingenuity is requiied 
to make work conform to the machine and to such 
cutters as we may have in stock. It is a good plan to 
make a trial piece for each gear, and to cut a few teeth 
in each trial piece to test the setting of the machine. 
Dummies or xhese trial pieces are called "dummies." If the gears 
are likely to be duplicated, each dummy can be marked 
and kept for future setting of the machine. Stamp all 
the data on the dummies ; it is better to spend a little 
time in marking dummies than a good deal of time 
hunting up, or trying to remember, old data. 

Let it be required to make two spiral gears to run 
with a ratio of 4 to 1, the distance between centers to 
be 3.125" (31"). 

By rule given in Chapter XII., Part I., we find the 
diameters of pitch circles will be 5" and 1;^". Let us 
take a spiral of 48" pitch for the large gear, and a 
spii'al of 12" pitch for the small gear. The circumfer- 
ence of the 5" pitch circle is 15.70796". Dividing 
the circumference by the pitch of the spiral, we have 
; 5-7^o^795 _ 32724-^ for tangent of angle of spiral. In 
the table the nearest angle to tangent, .32724", is 18° 10'. 
As before stated, the angle of the teeth in the small 
gear will be the same as the angle of teeth or spii'al in 



PROVIDENCE, R. I. 93 

the large gear. Now, tins rule gives the angle at the in'\ngieHriuop 
pitch surface only. Upon looking at a small screw of g"|J.aKIx)uVe8^ 
coarse pitch, it will be seen that the angle at bottom 
of the thread is not so great as the angle at top of 
thread ; that is, the thread at bottom is nearer parallel 
to the center line than that at the top. 




Fis.4r3 

This will be seen in Fig. 43, where A O is the center 
line ; h f shows direction of bottom of the thread, and 
d g shows dii'ection of top of thread. The angle Kfb 
is less than the angle A g d, This difference of angle 
is due to the warped nature of a screw thread, and 
sometimes makes it necessary to change the angle for 
setting work from the figured angle, when a rotary 
disk cutter is used, to prevent the cutter from marring 
the groove as the teeth of cutter enter and leave. 
How much to change the angle can be seen by inspec- 
tion when cutting the dummies. The change of angle 
will be more in a small gear of a given pitch than in a 
large gear of the same pitch. 

A rotary disk cutter is generally preferable, because Bisk-cutters. 
it cuts faster and holds its shape better. Yet it is 
hardly practical to cut low numbered pinions with 
rotary disk cutters, because for some distance below 
the pitch line the spaces are so nearly parallel. A part 
of the difficulty can be removed by making the cutter 
as small as is consistent with strength. Still more of 
the trouble can be done away with by making a cutter 
on a shank, the center of the work and the center of ^ simnk or End 

' _ _ Cutter. 

shank cutter then being in the same plane. When 
using a shank cutter the center of the work is perpen- 
dicular to the center of the cutter, no adjustment for 



94 JBROWN & SHARPE MFG. CO. 

angle being made. Strictly, a shank mill does not re- 
produce its own shape in cutting a spiral groove. In 
using a shank cutter, more care is necessary to see 
that the work does not slip. It may be well to rough 
out with a disk cutter and finish with a shank cutter. 
There is not generally much difficulty in involute or 
single-curve spii'al gears with disk cutters. 
c&hSSFon o^ ^ cylinder 2" diameter is to have spiral grooves 20° 
Pitch of Spiral, ^il-l^ the center line of cylinder; what will be the pitch 
of s^m-al? The circumference is 6.2832". The tan- 
gent of 20° is .36397. Dividing the circumference by 
the tangent of angle, w^e obtain ^;^^^^^ :=17.26" + for 
pitch of spiral. 

Before cutting into a blank it is well to make a 
slight trace of the spiral, with the cutter, after the 
machine is geared up, to see if the gears are properly 
arranged. Attention to this may avoid spoiling a, 
blank. 

The cutting of spiral gears develops some curious 
facts to one who may not have studied warped sur- 
faces. 

In the Universal Milling Machine we can cut a class 
of warped surfaces that will fit a straight edge in two 
directions. Thus, in Fig. 43, if it were possible to re- 
duce the diameter of screw and then cut the thread 
clear down to the center line A O, the bottom of the 
thread would be a straight line running through the 
center or the line A O itself. The sides would still be 
straight as in the figure. If we should cut a spiral 
groove with a plain rotary disk cutter, having parallel 
sides, the shape of the grooves would have but little 
resemblance to that of the cutter. Taking advantage 
of this principle, we learned the fact that spiral gears 
can be planed with a rack tool. 

Spiral Gears The fifcars, Fifif. 41, were planed. The tool was of 
cut with Eack fo ' & ' r 

Tool. the same shape as the spaces in the rack D D. All 

spiral gears of the same pitch could be planed with 
one tool. 

The natui'e of this can be seen when we consider 
that straight rack teeth can mesh with spiral gears, as 
in Fig. 40. 



PROVIDENCE, R. I. 95 

We have succeeded in cutting small spiral gears with 
a long fly tool, cutting on one side only. The shape 
of this fly tool was like a common lathe side tool. In 
this case, of course, the gears had to be reversed in 
order to finish both sides of teeth. 



96 



BROWN & SHARPE MFG. CO. 



CHAPTER VII. 

NORMAL PITCH OF SPIRAL GEARS— CURVATDRE OF PITCH 
SURFACE— FORM OF CUTTERS. 



Normal to a ^ Normal to a curve is a line perpendicular to the' 
tangent at the point of tangency. 




In Fig. 44, the line B C is tangent to the arc D E F^ 
and the line A E O, being perpendicular to the tan- 
gent at E, the point of tangency, is a normal to the 
arc. 

Fig. 45 is a representation of the pitch surface of a 
spiral gear. A' D' C is the circular pitch, as in Part 
I. A D C is the same circular pitch seen upon the 
periphery of a wheel. Let A D be a tooth and D C a 
space. Now, to make this space D C, the path of cut- 
ting is along the dotted line a h. By mere inspection, 
we can see that the shortest distance between two 
teeth along the pitch surface is not the distance 
ADC. 

Let the line A E B be perpendicular to the sides of 
teeth upon the pitch surface. A continuation of this 
line, perpendicular to all the teeth, is called the 
Norinal HelLc. The line A E B, reaching over a 
tooth and a space along the normal helix, is called the 
Normal Pitch. 



PROVIDENCE, R. I. 



97; 



\ 




Fi^. -io 



«98 BROWN & SHARPE MFG. CO. 

jTormai Pitch. The Normal Pitch of a spiral gear is then : The 
shortest distance between the centers of two consecutive 
teeth measured along the pitch surface. 

In spur gears the normal pitch and circular pitch 
are alike. In the rack D D, Fig. 40, the linear pitch 
and normal pitch are alike. 

Cutter for From the foregoing it will be seen that, if we should 
cut the space D C with a cutter, the thickness of which 
at the pitch line is equal to one-half the circular pitch, 
as in spur wheels, the space would be too wide, and 
the teeth would be too thin. Hence, spiral gears 
should be cut with thinner cutters than spur gears of 
the same circular pitch. 

The angle C A B is equal to the angle of the spiral. 
The line A E B corresponds to the cosine of the angle 
CAB. Hence the rule : Multiply the cosine of angle 

"^f p^^*h ^^^' ^f spiral by the circular pitch, and the product will be 
the normal pitch. One-half the normal pitch is the 
proper thickness of cutter at the pitch line. 

The addendum and working depth of tooth should 
correspond to the normal pitch, and not to the circular 
pitch. Thus, if the normal pitch is 12 diametral, the 
addendum should be -^^' •, the thickness .1309", and so 
on. The diameter of pitch circle of a spiral gear is 
calculated from the diametral pitch. Thus a gear of 
30 teeth 10 P would be 3" pitch diameter. 

But if the normal pitch is 12 diametral pitch, the 
blank will be ^^^' diameter instead of S^V • 

Normal Pitch It is evident that the normal pitch varies with the 
angle of spiral. The cutter should be for the normal 
pitch. In designing spiral gears, it is well to first 
look over list of cutters on hand, and see if there are 
cutters to which the gears can be made to conform. 
This may avoid the necessity of getting a new cutter, 
or of changing both drawing and gears after they are 
under way. To do this, the problem is worked the 
reverse of the foregoing ; that is : 

To make First calculate to the next finer pitch cutter than 

Angle of Spiral . ^ 

conform to Cut- would be required for the diametral pitch. 

ters given. 

Let us take, for example, a gear 10 pitch and 30 
teeth spiral. Let the next finer cutter be for 12 pitch 



vanes, 



PROVIDENCE, R. I. 99 

geai'S. The first thing is to find the angle that will 

make the normal pitch .2618", when the circular pitch 

is .3142". See table of tooth parts. This means (Fig. 

45) that the Ihie ADC will be .3142" when A E B is 

,2618". Dividing .2618" by .3142" (see Chapter III.), 

we obtain the cosine of the angle CAB, which is also 

the angle of the spiral, ^||^=.833. 

The same quotient comes by dividing 10 by 12. 

^|=:.833-|-. Looking in the table, we find the angle 

corresponding to the cosine .833 is 33° 30'. AYe now 

want to find the pitch of spiral that will give angle of 

33^° on the pitch surface of the wheel, 3" diameter. 

Dividing the circumference by the tangent of angle, 

we obtain the pitch of spiral (see Chapter V.) The 

circumference is 9.4248". The tangent of 33° 30' is 

.66188, ^^^^m|-=14.23; and we have for our spiral 

14.23" lead. 

When the machine is not arranged for the exact .'^hen exact 

^ Pitch cannot be 

pitch of sj)ii'al wanted, it is generally well enough to cut. 
take the next nearest spiral. A half of an inch more 
or less in a spiral 10" pitch or more would hardly be 
noticed in angle of teeth. It is generally better to 
take the next longer spiral and cut enough deeper to 
bring center distances right. When two gears of the 
same size are in mesh with their axes parallel, a change 
of angle of teeth or spiral makes no difference in the 
correct meshing of the teeth. 

But when gears of different size are in mesh, due ^^}}'^l Gears 
regard must be had to the spirals being in pitch, pro- sizes to Mesh. 
portional to their angular velocities (see Chapter YI.) 

We come now to the curvature of cutters for spiral 
gears; that is, their shape as to whether a cutter is 
made to ciit 12 teeth or 100 teeth. A cutter that is right, 
to cut a spiu' gear 3" diameter, may not be right for a stiapeof cut- 
spiral gear 3" diameter. To find the curvature of 
cutter, fit a templet to the blank along the line of the 
normal helix, as A E B, letting the templet reach over 
about two or three normal pitches. The curvature of 
this templet will be nearer a straight line than an arc 
of the addendum circle. Now find the diameter of a 
circle that will fit this templet, and consider this circle 



100 



BKO^^ & SHARPE MFG. CO. 



as the addendum circle of a gear for which we are to- 
select a cutter, reckoning the gear as of a pitch the 
same as the normal pitch. 




Fis. 4G 



Thus, in Fig. 46, suppose the templet fits a circle 
3^" diameter, if the normal pitch is 12 to inch, dia- 
metral, the cutter requii^ed is for 12 P and 40 teeth. 
The curvature of the templet will not be quite circular, 
but is sufficiently near for practical purposes. Strictly, 
a flat templet cannot be made to coincide with the 
normal helix for any distance whatever, but any greater 
refinement than we have suggested can hardly be 
carried out in a workshop. 

The difi'erence between normal pitch and linear or 
circular pitch is plainly seen in Figs. 40 and 41. 

The rack D D, Fig. 40, is of regular form, the depth, 
of teeth being ||- of the circular pitch, nearly (.6866 of 
the pitch, accurately). If a section of a tooth in either 
of the gears be made square across the tooth, that is a 
normal section , the depth of the tooth will have the 
same relation to the thickness of the tooth as in the 
rack just named. 

But the teeth of spii'al gears, looking at them upon 
the side of the gears, are thicker in proportion to their 
depth, as in Fig. 41. This difference is seen between 



PROVIDENCE, R. I. 101 

the teeth of the Uyo racks D D and E E, Fig. 40. In 
the rack D D we have 20 teeth, while in the rack E E 
we have but 14 teeth ; yet each rack will run with each 
of the spiral gears A, B or C, Fig. 41, but at different 
angles. 

The teeth of one rack will accurately fit the teeth of 
the other rack face to face, but the sides of one rack 
will then be at an angle of 45° with the sides of the 
other rack. At F is a guide for holding a rack in mesh 
with a gear. 



102 BROWN & SHARPE MFG. CO. 



CHAPTER VIII. 
SCREW GEARS AHD SPIRAL GEARS— GENERAL REMARKS. 



Sph-ai^Gears ^^ '^^^ working of Spiral gears is generally smootliGr 
than spur gears. A tooth does not strike along its 
whole face or length at once. Tooth contact first takes 
place at one side of the gear, passes across the face 
and ceases at the other side of the gear. This actio?! 
tends to cover defects in shape of teeth and the adjust- 
ment of centers. 

Since the invention of machines for producing accu- 
rate epicyloidal and involute curves, it has not so often 
been found necessary to resort to spiral gears for 
smoothness of action. A greater range can be had in 
the adjustment of centers in spiral gears than in spur 
gears. The angle of the teeth should be enough, so 
that one pair of teeth will not part contact at one side 
of the gears until the next pair of teeth have met on the 
other side of the gears. When this is done the gears 
will be in mesh so long as the circumferences of their 
addendum circles intersect each other. This is some- 
times necessary in roll gears. 

Relative to spur and bevel gears in Part I., Chapter 
XXL, it was stated that all gears finally wore them- 
selves out of shape and might become noisy. Spiral 
gears may be worn out of shaj^e, but the smoothness 
of action can hardly be impaired so long as there are 
any teeth left. For every quantity of wear, of course, 
there will be an equal quantity of backlash, so that if 
gears have to be reversed the lost motion in spiral 
gears will be as much as in any gears, and may be 
Ti^n^shlffs^of i^^^i'^ if there is end play of the shafts. In spiral gears 
Spiral Gears, ^j^ere is end pressure upon the shafts, because of the 
screw-like action of the teeth. This end pressure is 
sometimes balanced by putting two gears upon each 
shaft, one of right and one of left hand spii'al. 



PKOVIDENCE, R. I. 103 

The same result is obtained in solid cast gears by 
making the pattern in two parts — one right and one 
left-hand spiral. Such gears are colloquially called 
"herring-bone gears/' 

In an internal spiral gear and its pinion, the spirals 
of both wheels are either right-handed or left-handed. 
Such a combination would hardly be a mercantile 
product, although interesting as mechanical feat. 

In screw or worm-gears the axes are generally at 
right angles, or nearly so. The distinctive features of 
screw gearing may be stated as follows : 

The relative angular velocities do not depend upon 
the diameters of pitch- cylinders, as in Chapter I., 
Part I. Thus the worm in Chapter XL, Fig. 18, can^ Distinctive 
be any diameter — one inch or ten inches — without Screw Gearing, 
affecting the velocity" of the worm-wheel. The direc- 
tion in which a worm-wheel turns depends upon 
whether the worm has a right-hand or left-hand thread. 
When angles of axes of worm and worm-wheel are 
oblique, there is a practical limit to the directional 
relation of the worm-wheel. The rotation of the 
worm-wheel is made by the end movement of the 
worm-thread. 

The term worm and worm-wheel, or worm-gearing, 
is applied to cases where the worms are cut in a lathe. 

If we let two cylinders touch each other, their axes 
be at right angles, the rotation of one cylinder will 
have no tendency to turn the other cylinder, as in 
Chapter I., Part I. 

We can now see why worms and worm-w^heels wear ^^''^y worm 

*' Wheels wear 

out faster than other gearing. The length of worm- so last, 
thread, equal to more than the entu'e circumference of 
worm, comes in sliding contact with each tooth of the 
wheel during one turn of the wheel. 

The angle of a worm-thread can be calculated the 
same as the angle of teeth of spiral gear. 



104 BROWX & SHAKPE MFG. CO. 



CHAPTER IX. 

CONTINUED FRACTIONS— SOME APPLICATIONS IN MACHINE 
CONSTRUCTION. 



Deflnition of j^ continued fraction is one which has unity for its 

a Continued _ -^ 

Traction. numerator, and for its denominator an entire number 
plus a fraction, which fraction has also unity for its 
numerator, and for its denominator an entire number 
plus a fraction, and thus in order. 
The expression, '^ . 

4 + l_ 

^ is called a continued frac- 
tion. By the use of continued fractions, we are ena- 
Practicai use l,]ed to find a fraction expressed in smaller numbers, 

of Continued ^ i zv • i 

Fractions. that, for practical purposes, may be sufficiently near m 
value to another fraction expressed in large numbers. 
If W3 were required to cut a worm that would mesh 
with a gear 4 diametral pitch (4 P.), in a lathe having 

3 to 1-inch linear leading screw, we might, without 
continued fractions, have trouble in finding change 
gears, because the circular pitch corresponding to 

4 diametral pitch is expressed in large numbers : 

A P — .tJLSjJ^ P' 
^-^ — 10000 -^ • 

This example will be considered farther on. For 
illustration, we will take a simpler example. 

What fraction expressed in smaller numbers is near- 
est in value to yVc^ Dividing the numerator and the 
denominator of a fraction by the same number does 
not change the value of the fraction. Dividing both 
o n%Tn u e'd ^^^^^ of ^j\ by 29, we have ^TT or, what is the 
same thing expressed as a continued fraction, s+HT. The 
continued fraction 5 4-_i_ is exactly equal to -f_9_, if 
now, we reject the ^V? ^^^ fraction \ will be larger 



C o n 1 1 n u 
Fractions, 



PROVIDENCE, R. I. 

than 5-t-^t,, because tlie denominator has been dimin- 
ished, 5 being- less than So^.j. l is the nearest fraction 
to y2/(j- expressed in sma ler numbers than 29 for a 
numerator and 146 for a denominator. Eeducing ^ 
and ^^\ to a common denominator, we have l^-i^J 
and iVg-^t"! 0- Subtracting one from the other, we 
have yjo, which is the difference between V and -f^^. 
Thus, in thinking of j"/g as t, we have a pretty good 
idea of its value. 

In the French metric system, a millimetre is equal 
to .03937 inch ; what fraction in smaller terms ex- 
presses .03937" nearl}^? .03937, in a vulgar fraction, 
is y-o-||-oi)-. Dividing both numerator and denominator 
by 3937, we have 2.51I-14. Bejecting from the de- 
nominator of the new fraction, 3 jf^y, the fraction ^^3 
gives us a pretty good idea of the value of .03937". 
If in the expression, 25+1544, we divide both terms of 
the fraction -Jf 54 by 1575, the value will not be changed. 
Performing the division, we have ^ 

25 + 1 



10/ 



2 + 787 
1575- 

We can now divide both terms of ^A?^ by 787, 
without changing its value, and then substitute the 
new fraction for -^'^g^j in the continued fraction. 

Dividing again, and substituting, we have : 







25 + 1 








2 + 1 








2+ 1 
787 




as the 


continued 


fraction that is 


exactly equal to 


.03937. 








In performing \ 


:he divisions, the ^ 


work stands thus : 




393'; 


") 100000 (23 

7874 

21260 

19685 

1575) 3937 (2 

3150 

787) 1575 (2 
1574 








1) 787 (787 

787 



•0- 
That is, dividing the last divisor by the last remain- 
der, as in finding the greatest common divisor. The 



106 



BROWN & SHARPE MFG. CO. 

quotients become the denominators of the continued 
fraction, with unity for numerators. The denominators 
25, 2, and so on, are called incomplete quotients, since 
they are only the entire parts of each quotient. The 
first expression in the continued fraction is ^^ or 
.04 — a little larger than .03937. If, now, we take 
25-qrT, w^e shall come still nearer .3937. The expres- 
sion 25^T is merely stating that 1 is to be divided by 
25|-. To divide, we first reduce 254- to an improper 
fraction, ^, and the expression becomes 2T, or one 
divided by ^. To divide by a fraction, "Invert the 
divisor, and proceed as in multiplication." We 
then have -fj as the next nearest fraction to .03937. 
-g2_=.0392 + , which is smaller than .03937. To get still 
nearer, we take in the next part of the continued frac- 
tion, and have 1 



2 + 1 

2' 

We can bring the value of this expression into a 
fraction, with only one number for its numerator and 
one number for its denominator, by performing the 
operations indicated, step by step, commencing at the 
last part of the continued fraction. Thus, 2 4-|-, or 
2|^, is equal to |, Stopping here, the continued frac- 
tion would become 1 

25+J_ 

2- 

1 \ 

Now, ^ equals f , and we have 25 + 2^. 25f equals. 

2 5 

^1^ ; substituting again, we have j_Ji. Dividing 1 by 

5 
■3^^, we have yfy. yfy is the nearest fraction to 

.03937, unless we reduce the whole continued fraction 
1^ 

25+ 1 

2 + 1 

^ + 1 , which would give us back the .03937 itself. 

787 

^|^=. 03937007, which is only ^^^i_^^ larger 

.03937. It is not often that an approximation will 

come as near as this. 

Practical use This ratio, 5 to 127, is used in cutting millimeter 

Example. thread screws. If the leading screw of the lathe was. 



PROVIDENCE, R. I. 107 

1 to one inch, the change gears would have the ratio of 
5 to 127; if 8 to one inch, the ratio would be 8 times 
as large, or 40 to 127; so that with leading screw 8 to 
inch, and change gears 40 and 127, we can cut milli- 
meter threads near enough for practical purposes. 

The foregoing operations are more tedious in de- 
scription than in use. The steps have been carefully 
noted, so that the reason for each step could be seen 
from rules of common arithmetic, the operations being 
merely reducing complex fractions. The reductions, 
■gV' 'iv tIt' ^^^-i ^^'® called conver gents, because they 
come nearer and nearer to the required .03937. The 
operations can be shortened as follows: 

Let us find the fractions converging towards .7854", Example; 
the circular pitch of 4 diametral pitch, .7854=y\fo^\j*^; 
reducing to lowest terms, we have ||!f J. Applying 
the operation for the greatest common divisor: 

3927) 5000 (1 
3927 

1073) 3927 <3 
3219 
708) 1073 (1 
_708 
365) 708 (1 
3[)5 

843) 365 (1 
343 

22) 343 (15 
22 
123 

no 

13) 22 (1 
13 

9) 13(1 
9 

4) 9 (2 
8 

1) 4 (4 
4 


Bringing the various incomplete quotients as de- 
nominators in a continued fraction as before, we have : 

1 

1 + 1 

3 + 1 

1 + 1 

1 + 1 

15 + 1 

1 + 1 

1 + 1 

2 + i 



108 BROWN & SHARPE MFG. CO 

Now arrange each partial quotient in a line, thus : 
13111 15 1 1 2 4 

1 3 4 1 11 1T2 183 3 51 893 3927 
-■- 4 "5 ¥ T4 2T9" "23-3 TT2 TT3T loWO 

Now place under the first incomplete quotient the 
first reduction or convergent ^, which, of course, is 1 ; 
put under the next partial quotient the next reduction or 
convergent j— r or ^r, which becomes f . 

1 is larger than .7854, and f is less than .7854. 

Having made two reductions, as previously shown, 
we can shorten the operations by the following rule 
for next convergents : Multiply the numerator of the 
convergent just found by the denominator of the next 
term of the continued fraction, or the next incomplete 
quotient, and add to the product the numerator of the 
preceding convergent ; the sum will be the numerator 
of the next convergent. 

Proceed in the same way for the denominator. 
Continue until the last convergent is the original frac- 
tion. Under each incomplete quotient or denominator 
from the continued fraction arranged in line, will be 
seen the corresponding convergent or reduction. The 
convergent ^^ is the one commonly used in cutting 
racks 4 P. The equivalent decimal to ^J is .7857 + ) 
being about yo^fo~o large. In three settings for rack 
teeth, this error would amount to about .001". 

For a worm, this corresponds to ^^ threads to 1" ; 
now-, with a leading screw of lathe 3 to 1", we would 
want gears on the spindle and screw in a ratio of 33 
to 14. 

Hence, a gear on the spindle with 66 teeth, and a 
gear on the 3 thread screw of 28 teeth, would enable 
us to cut a worm to fit a 4 P gear. 



PROVIDENCE, R. 



109 



CHAPTER X. 
♦ ANGLE OF PRESSURE 



lu Fig. 47, let A be any flat disk lying upon a hori- 
zontal plane. Take any piece, B, with a square end, 
<x h. Press against A with the piece B in the direction 
of the arrow. 





Fig. 4' 



Fio;. 48 



It is evident A will tend to move directly ahead of 
B in the normal line c d. Now (Fig. 48) let the piece 
B, at one corner^/*, touch the piece A. Move the 2)iece 
B along the line d e, in the direction of the arrow. 

It is evident that A will not now tc^nd to move in 

the line d e, but will tend to move in the direction of 

the normal c d. When one piece, not attached, presses 

against another, the tendency to move the second 

piece is in the direction of the normal, at the point of 

contact. This normal is called the line of pressure. i^ineofPn 

>■' ' ure. 

The angle that this line makes with the path of the 
impelling piece, is called the angle of pressure. 

In Part I., Chapter IV., the lines B A and B A' are 
called lines of pressui'e. This means that if the gear 



110 BROWN & SHARPE MFG. CO. 

drives the rack, the tendency to move the rack is not 
in the direction of pitch line of rack, but either in the 
direction B A or B A', as Ave turn the wheel to the left 
or to the right. 

The same law holds if the rack is moved in the 
direction of the pitch line ; the tendency to move the 
wheel is not directly tangent to the pitch circle, as if 
driven by a belt, but in the direction of the line of 
pressure. Of course the rack and wheel do move in 
the paths prescribed by their connections with the 
framework, the wheel turning about its axis and the 
rack moving along its ways. This pressure, not in a 
direct path of the moving piece, causes extra friction 
in all toothed gearing that cannot well be avoided. 

Although this pressure works out by the diagram, 
as we have shown, yet, in the actual gears, it is not at 
all certain that they will follow the law as stated, 
because of the friction of teeth among themselves. If 
the driver in a train of gears has no bearing upon its 
tooth-flank, we apprehend there will be but little 
tendency to press the shafts apart. 
Arc of Action. fjj^g ^^^ through whicli a wheel passes while one of 

its teeth is in contact is called the arc of action. 
tenf^of ° imer" ^^i^til within a few years, the base of a system of 
^^^^J s ® ^ '^ 1 ® double- curve interchangeable gears was 12 teeth. It 
is now 15 teeth in the best practice (see Chapter VII., 
Part I.) 

The reason for this change was : the base, 15 teeth, 
gives less angle of pressure and longer arc of contact,. 
and hence longer lifetime of gears. 



PRO^^DENCE, R. T. HI 



CHAPTER XT. 
IHTERHAL GEARS— TABLES 



In Part I., Chapter YIII., it was stated that the 
space of an internal gear is the same as the tooth of a 
spur gear. This applies to involute or single-curve 
gears as well as to double-curve gears. 

The sides of teeth in involute internal gears will be 
hollowing. It, however, has been customary to cut 
internal gears with spur gear-cutters, a No. 1 cutter 
generally being used. This makes the teeth sides 
convex. Special cutters should be made for coarse Special Cut, 

., T -, ■, -, TT---i 1 t^^'3 for coarse 

pitch double-curve gears, in designmg mternal gears, Pitch. 
it is sometimes necessary to depart from the system 
with 15-tooth base, so as to have the pinion differ from 
the wheel by less than 15 teeth. The rules given in 
Part I., Chapters YII. and YIII., will apply in making 
gears on any base besides 15 teeth. If the base is 
low-numbered and the pinion is small, it may be neces- 
sary to resort to the method gi-ven at the end of Chaj)- 
ter YII., because the teeth may be too much rounded 
at the points by following the approximate rules. 
The base must be as small as the dliference between Base for in- 

, ., . . rr^i 1 ternal Gear 

the internal gear and its pinion. The base can be Teeth, 
smaller if desired. 

Let it be required to make an internal gear, and 
pinion 24 and 18 teeth, 3 P. Here the base cannot 
be more than 6 teeth. 

In Fig. 49 the base is 6 teeth. The arcs A K and 
O k, drawn about T, have a radius equal to the radius 
of the pitch circle of a 6-tooth gear, 3 P, instead of a 
15-tooth gear, as in Chapter YIII., Part I. 

The outline of teeth of both gears and pinion is T^^s^'iption of 
made similar to the gear in Chapter YIII. The same 



112 



BROWN & SHARPE MFG. CO. 




P = 3 

N =24 and 1 

P'= 1.0472" 

t- 5236° 

=. .3333' 

D-= .6666 

S+f= .3857 

P"+/= .7190' 



NTERNAL GEAR AND PINION IN MESH. 



PROVIDENCE, R. I. IIB" 

letters refer to similar parts. The clearance circle is, 
however, drawii on the outside for the internal gear. 
As before stated, the spaces of a spur wheel become 
the teeth of an internal wheel. The teeth of internal 
gears require but little for fillets at the roots ; they 
are generally strong enough without fillets. The 
teeth of the pinion are also similar to the gear in 
Chapter VIIL, substituting G-tooth for 15-tooth base. 
To avoid confusion, it is well to make a complete 
sketch of one gear before making the other. The arc 
of action is longer in internal gears than in external 
gears. This property sometimes makes it necessary 
to give less fillets than in external gears. 

In Fig. 49 the angle K T A is 30^ instead of 12°, as 
in Fig. 12. This brings the line of pressure L P at 
an angle of 60° wdth the radius C T, instead of 78°. 
A system of spur gears could be made upon this 
6-tooth base. These gears would interchange, but no 
gear of this 6-tooth system would mesh witli a double- 
curve gear made upon the 15-tooth system in Part 1. 

The "Willis" double-curve Odontograph is based Baseofwinis, 

° ^ Odontograpli. 

upon radial flank gears of 1 2 teeth. 



114 



BROWN & SHARPE MFG. CO. 



TABLE OF DECIMAL EQUIVALENTS 

OF 8ths, 16ths, 32nds and 64Tfls of an inch. 



8ths. 

J=.125 
i=.250 

^=.500 
t==.625 
f=.750 

i=.815 

IGths. 

yV=.0625 
j\=.1815 

A=.4375 

yV=.5625 
ii = .6875 
if =.8125 
if =.9375 

32nds. 

31-2 = . 03125 
/^=. 09375 
3^=. 15625 
3V= . 21875 



3^=. 28125 
11=. 34375 
i|=. 40625 
If =.46875 
i;i=. 53125 
-P= . 59375 
2 1-=. 65625 
||=. 71875 
||=. 78125 
|4=. 84375 
||=. 90625 
|i=. 96875 



64ths. 

^3 = . 015625 

g3^=. 046875 
/^=. 078125 
-g5^=. 109375 
/j=. 140625 
-11=. 171875 
If =.203125 
If =.234375 
f|=. 265625 



J-|=. 296875 
|i=. 328125 
If =.359375 
If =.390625 
If =.421875 
||=. 453125 
fi=. 484375 
If =.515625 
f}=. 546875 
II = .578125 
||=. 609375 
|i= . 640625 
f|=. 671875 
If =.703125 
f f = 734375 
If = . 765625 
fi=. 796875 
If =.828125 
If =.859375 
||=. 890625 
f|-=. 921875 
|i=. 953125 
If =.984375 



PROVIDENCE, R. I. 



115 



TABLE OF DECIMAL EQUIVALENTS 



OF MILLIMETERS AND FRACTIONS OF MILLIMETERS. 



mm. Inches. 

00079 
00157 
00236 
00315 
00394 
00472 
00551 
00630 
00709 
00787 
00866 
00945 
01024 
01102 
01181 
01260 
01339 
01417 
01496 
01575 
01654 
01732 
01811 
01890 
01969 



-zV 
-eV- 

_5_- 
5 0" 

/o = 

-ir 

a-- 

1 3- 

u= 

U'- 

-w 

1 9- 

2 0- 
^0- 
2J- 
50" 
2 2- 
■^77- 
23- 

U-- 



mm. Inches. 

If =.02047 

02126 

, 02205 

02283 

, 02362 

,02441 

02520 

, 02598 

,02677 

. 02756 

02835 

02913 

02992 

03071 

03150 

03228 

03307 

03386 

03465 

03543 

03622 

03701 

03780 

03858 

03937 



mm. Inches. 



■n-- 

29- 

U-- 

M= 

3 3.- 
§0 

l§ = 

l*= 

3 .5 - 

To- 

3 6- 

To- 

31- 

U-- 

ii-- 

n-- 
U-- 

4 2- 

5 0" 
4ji- 
50" 

U-- 

46- 
50 - 
Al- 
n-- 

4 9- 

To- 

1:: 



9: 
10: 
11: 
12: 
13: 
14: 
15: 
16: 
17: 
18: 
19: 
20: 
21: 
22: 
23: 
24: 
r>5: 
26: 



: .07874 

: .11811 

: .15748 

: .19685 

: .23622 

: .27559 

: .31496 

: .35433 

: .39370 

: .43307 

: .47244 

: .51181 

: .55118 

: .59055 

: .62992 

: .06929 

: .70866 

: .74803 

: .78740 

: .82677 

: .86614 

: .90551 

: .94488 

: .98425 
1.02362 



10 mm. = 1 Centimetcr=:0.3937 inches. 
10 cm. =1 Decimeter =3.937 
10 dm. =1 Meter =39.37 

25.4 mm.=l English Inch. 



116 



BROWN & SHARPE MFG. CO. 



NATUKAL SINE. 



Be?. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 







.00000 


.00291 


.00581 


.00872 


.01168 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02326 


.02617 


.02908 


.08199 


.03489 


88 


2 


.03489 


.03780 


.04071 


.04361 


.04652 


.04943 


.05233 


87 


3 


.052S3 


.05524 


.05814 


.06104 


.06395 


.06685 


.06975 


86 


4 


.06075 


.07265 


.07555 


.07845 


.08135 


.08425 


.08715 


85 


5 


.08715 


.09005 


.09295 


.09584 


.09874 


.10168 


. 10452 


84 


6 


.10452 


.10742 


.11031 


.11820 


.11609 


.11898 


.12186 


83 


7 


.12180 


.12475 


.12764 


.13052 


.13341 


. 13629 


.13917 


82 


8 


.13917 


. 142U5 


. 14493 


.14780 


.15068 


.15356 


.15643 


81 


9 


. 15643 


.15930 


.16217 


.16504 


.16791 


.17078 


. 17364 


80 


10 


.17364 


.17651 


.17937 


.18223 


.18509 


.18795 


.19080 


79 


11 


.19080 


.19366 


.19651 


.19936 


.20221 


.20506 


.20791 


78 


12 


.2D791 


.21075 


.21359 


.21644 


.21927 


.22211 


.22495 


1 77 


13 


.22495 


.22778 


.23061 


.23344 


.23627 


.23909 


.24192 


76 


14 


.24192 


.24474 


.24756 


.25038 


.25319 


.25600 


.25881 


75 


15 


.25881 


.26162 


.26443 


.26723 


.27004 


.27284 


.27563 


74 


16 


.27563 


.27848 


.28122 


.28401 


.28680 


.28958 


.29237 


! 73 


17 


.29237 


.29515 


.29793 


.30070 


.30847 


.80624 


.30901 


72 


18 


.30901 


.81178 


.31454 


.31733 


.32006 


.32281 


.32556 


i "^1 


19 


.33556 


.32831 


.33106 


.33380 


.33654 


.33928 


.34202 


70 


20 


.34202 


.34475 


.34748 


.35020 


.85293 


.35565 


.35836 


1 69 


21 


.35886 


.36108 


.36379 


.36650 


.36920 


.37190 


.37460 


68 


22 


.87460 


.37730 


.37999 


.38268 


.38536 


.38805 


.89073 


67 


23 


.39073 


.39340 


.39607 


.39874 


.40141 


.40407 


.40673 


66 


24 


.40073 


.409C9 


.41204 


.41469 


.41733 


.41998 


.42261 


65 


25 


.42261 


.42525 


.42788 


.43051 


.43313 


.43575 


.48837 


64 


26 


.43837 


.44098 


.44359 


.44619 


.44879 


.45139 


.45399 


63 


27 


.45399 


.45658 


.45916 


.46174 


.46432 


.46690 


.46947 


62 


28 


.46947 


.47203 


.47460 


.47715 


.47971 


.48226 


.48481 


61 


29 


.48481 


.48785 


.43989 


.49242 


.49495 


. 49747 


.50000 


00 


30 


.50000 


.50251 


. 50503 


.50758 


.51004 


.51254 


.51503 


59 


31 


.51503 


.51752 


.52001 


, .52249 


.52497 


.52745 


.52991 


58 


82 


.52991 


.53208 


.53484 


.58730 


.53975 


.54219 


.54463 


57 


33 


.54463 


.54707 


. 54950 


.55193 


.55436 


.55677 


.55919 


56 


34 


.55919 


.56160 


.56400 


.56640 


.56880 


.57119 


.57357 


55 


85 


.57357 


.57595 


.57833 


.58070 


.58806 


.58542 


.58778 


54 


86 


.58778 


.59013 


.59243 


.59482 


.59715 


.59948 


.60181 


53 


87 


.60181 


.60413 


.60645 


. 60876 


.61106 


.61336 


.61566 


52 


88 


.61566 


.61795 


.62028 


.62251 


.62478 


.62705 


.62932 


51 


39 


.62932 


.68157 


.63383 


.68607 


.68882 


.64055 


.64278 


1 50 


40 


.64278 


.64501 


.64723 


.64944 


.65165 


.65386 


.65605 


1 49 


41 


.65605 


.65825 


.66043 


.66262 


.66479 


.66696 


.66913 


i 48 


42 


.66918 


.67128 


.67344 


.67559 


.67773 


.67986 


.68199 


; 47 


43 


.68199 


.68412 


.68624 


.68885 


.69046 


.69256 


.69465 


! 46 


44 


.69465 


.69674 


.69888 


. 70090 


.70298 


.70504 


.70710 


45 




60' 


50' 


40' 


- 


20' 


10' 


0' 


I>eg. 



NATUKAL COSINE. 



PROVIDENCE, R. I. 



117 



NATURAL SINE. 



Beg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 




45 


.70710 


: .70916 


.71120 


.71325 


.71528 


.71731 


.71934 


44 


46 


.71934 


i .72135 


.72330 


; .72537 


.72737 


.72930 


.73135 


' 43 


47 


.73135 


.73333 


73530 


.73727 


. 73923 


.74119 


.74314 


' 42 


48 


.74314 


.74508 


.74702 


.74895 


. 75088 


.75279 


.75471 


41 


49 


.75471 


.75661 


. 75851 


. 76040 


. 70229 


.70417 


. 70004 


40 


50 


.76J04 


. 76791 


.70977 


.77102 


.77347 


.77531 


.77714 


39 


51 


.77714 


.77897 


.78079 


. 78260 


.78441 


.78021 


. 78801 


38 


52 


.78801 


. 78979 


.79157 


.79335 


.79512 


.79088 


. 79803 


37 


53 


.79863 


. 80038 


.80212 


. 80385 


. 80558 


.80730 


.80901 


30 


54 


.80901 


.81072 


.81242 


.81411 


.81580 


.81748 


.81915 


35 


55 


.81915 


.82081 


.82247 


.82412 


.82577 


.82740 


.82903 


34 


56 


.82903 


.83060 


.83227 


.83383 


.83548 


.83708 


.83807 


33 


57 


.83867 


.84025 


.84182 


.84339 


.84495 


. 84050 


.84804 


1 32 


' 58 , 


.84804 


.84958 


.85111 


.85264 


.85415 


. 85500 


.85710 


31 


59 


.85716 


.85860 


.80014 


.80102 


.83310 


.80450 


.80002 


30 


60 


.86602 


.80747 


.80892 


.87035 


.87178 


.87320 


.87402 


29 


01 1 


.87462 


.87602 


. 87742 


.87881 


.88020 


.88157 


.88294 


28 


62 ! 


.88294 


.88430 


.88500 


.88701 


. 88835 


.88908 


.89100 


27 


03 


.89100 


.89232 


.89303 


.89493 


.89022 


.89751 


.89879 , 


20 


64 


.89879 


.90000 


.90132 


.90258 


.90383 


.90507 


.90030 


25 


05 


.90630 


.90753 


.90875 


.90990 


.91110 


.91235 


.91354 i 


24 


66 : 


.91354 


.91472 


.91589 


.91700 


.91821 


.91930 


.92050 : 


23 


07 i 


.92050 


.92103 


.92270 


. 92388 


.92498 


.92009 


.92718 


22 


08 1 


.92718 


.92827 


.92934 


.93041 


.93148 


.93253 


.93358 ! 


21 


09 


.93358 


.93401 


.93505 


.93007 


.93708 


.93809 


.93909 i 


20 


70 


.93969 


.94008 


.94100 


.94204 


.94300 


. 94450 


.94551 1 


19 


71 


.94551 


.94640 


.94739 


.94832 


.94924 


.95015 


.95105 


18 


72 


.95105 


.95195 


.95283 


.95371 


.95458 


.95545 


.95030 


17 


73 


.95630 


.95715 


.95799 


.95882 


.95904 


.90045 


.90120 


10 


74 


.96126 


.90205 


.90284 


.90303 


.90440 


.90510 


.90592 


15 


75 


.96592 


.90007 


.90741 


.90814 


.90887 


.90958 


.97029 


14 


76 


.97029 


.97099 


.97108 


.97237 


. 97304 


.97371 


.97437 


13 


77 


.97437 


.97502 


.97500 


.9702!) 


.97092 


.97753 


.97814 


12 


78 


.97814 


.97874 


.97934 


.97992 


.98050 


.98100 


.98102 


11 


79 


.98162 


.98217 


.98272 


.98325 


.98378 


.98429 


.98480 


10 


80 


.98480 


.98530 


.98580 


.98028 


.98070 


. 98722 


.98708 


9 


81 1 


.98768 


.98813 


.98858 


.98901 


.98944 


.98985 


.99020 


8 


82 ' 


.99026 


.99000 


.09106 


.99144 


.99182 


.99218 


.99254 


7 


83 


.99254 


.99289 


.99323 ! 


.99357 


.99389 


.99421 


.99452 





84 


.99452 


.99482 


.99511 ! 


.99539 


.99507 


.99593 


.99019 


5 


85 


.99619 


.99044 


.99668 1 


.99091 


.99714 


.99735 


.99750 


4 


86 


.99756 


.99770 


.99795 1 


99813 


.99830 


.99847 


.99803 


3 


87 


.99863 


.99877 


.99891 


. 99904 


.99917 


.99928 


.99939 


2 


88 


.99939 


.99948 


.99957 


.99905 


. 99972 


.99979 


.99984 


1 


89 1 


.99984 


.99989 


.99993 


.99990 


.99998 


.99999 


1.0000 





1 


60' 


50' 


40' j 


30' 


20' 


10' 


c 


Deg. 



NATURAL COSINE. 



118 



BROWN & SHARPE MFG. CO. 



NATURAL TANGENT. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' 


! 
t 





.00000 


.00390 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.03036 


.02327 


.02618 


.03909 


.03300 


.03493 


88 


2 


.03493 


.03783 


.04074 


.04366 


.04657 


.04949 


.05340 


87 


3 


.05240 


.05532 


.05834 


.06116 


.06408 


.06700 


.06993 


86 : 


4 


.06993 


.07285 


.07577 


.07870 


.08163 


.08455 


.08748 


85 


5 


.08748 


.09042 


.09335 


.09628 


.09933 


.10316 


.10510 


84 


6 


.10510 


. 10804 


.11099 


.11393 


.11688 


.11983 


.13378 


83 ; 


7 


.12378 


.12573 


.12869 


.13165 


.13461 


. 13757 


.14054 


83 


8 


.14054 


.14350 


.14647 


.14945 


.15343 


.15540 


.15838 


81 


9 


.15838 


.16136 


.16435 


.16734 


.17033 


.17333 


.17633 


80 


10 


.17632 


.17932 


.18233 


.18533 


.18834 


.19136 


.19438 


79 


11 


.19438 


. 19740 


.20043 


.20345 


.30648 


.30951 


.31355 


78 ; 


13 


.31355 


.21559 


.21864 


.22169 


.33474 


.33780 


.33086 


77 


13 


.33086 


.23393 


.23700 


.24007 


.34315 


.34634 


.34933 


76 


14 


.34933 


.25242 


.25551 


.25861 


.36173 


.36483 


.36794 


75 


15 


.36794 


.27106 


.37419 


.27733 


.38046 


.38360 


.38674 


74 


16 


.38674 


.28989 


.39805 


.29631 


.39938 


.30355 


.30578 


73 


17 


.30573 


.30891 


.31210 


.31539 


.31850 


.33170 


. 33493 


73 


18 ' 


.33493 


.32813 


.33130 


.33459 


.33783 


.34107 


.34433 


71 


19 


.34433 


.34758 


.35084 


.35411 


.35739 


.36067 


.36397 


70 


20 


.36397 


.36726 


.37057 


.37388 


.37730 


.38053 


.38386 


69 ' 


21 


.38386 


.38720 


.39055 


.39391 


.39737 


.40064 


.40403 


68 


23 ' 


.40403 


.40741 


.41080 


.41431 


.41763 


.43104 


.43447 


67 


23 


.43447 


.42791 


.43135 


.43481 


.43837 


.44174 


.44533 


66 


24 


.44523 


.44871 


.45321 


.45573 


.45934 


.46377 


.46630 


65 


25 


.46630 


.46985 


.47341 


.47697 


.48055 


.48413 


.48773 


64 


26 


.48773 


.49133 


.49495 


.49858 


.50331 


.50586 


.50952 


63 


27 ' 


.50953 


.51319 


.51687 


.53056 


.53437 


.53798 


.53170 


63 


28 i 


.53170 


.58544 


.53919 


.54295 


.54673 


.55051 


.55430 


61 


29 


.55430 


.55811 


.56193 


.56577 


.56961 


.57347 


.57735 


60 


30 


.57735 


.58123 


.58513 


.58904 


.59397 


.59690 


.60086 


59 


31 


.60086 


.60482 


.60880 


.61380 


.61680 


.63083 


.63486 


58 


33 


.63486 


.62892 


.63398 


.63707 


.64116 


.64538 


.64940 


57 ; 


33 : 


.64940 


.65355 


.65771 


.66188 


.66607 


.67038 


.67450 


56 


34 


.67450 


.67874 


.68300 


.68728 


.69157 


.69588 


.70030 


55 • 


35 


.70020 


.70455 


.70891 


.71339 


.71769 


.73310 


.73654 


54 


36 


.73654 


.73099 


.73546 


.73996 


.74447 


.74900 


. 75355 


53 , 


37 


.75355 


.75812 


.76371 


.76733 


.77195 


.77661 


.78138 


53 


38 


.78138 


.78598 


.79069 


.79543 


.80019 


.80497 


80978 


51 


39 


.80978 


.81461 


.81940 


.83433 


.83933 


.83415 


.83910 


50 


40 


.83910 


.84400 


.84900 


.85408 


.85913 


.86419 


.86928 


49 


41 


.86928 


.87440 


87955 


.88473 


.88993 


.89515 


.90040 


48 


42 


.90040 


.90568 


.91099 


.91633 


.92169 


.93709 


.93351 


47 


43 


.93351 


.93796 


.94345 


.94896 


.95450 


.96008 


.96568 


46 


44 


.96568 


.97133 


.97699 


.98369 


.98843 


.99419 


1.0000 


45 




60' 


50' 


40' 


30' 


20' 


10' 


C 


Deg. 



NATURAL COTANGENT. 



PROVIDENCE, R. I. 



119 



NATURAL TANGENT. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


60 




45 


1.0000 


1.0058 


1.0117 


1.0176 


1.0235 


1.0295 


1.0355 


44 


46 


1.0355 


1.0415 


1.0470 


1.0537 


1.0599 


1.0661 


1.0723 


43 


47 


1.0723 


1.0786 


1.0849 


1.0913 


1.0977 


1.1041 


1.1106 


42 


48 , 


1.1106 


1.1171 


1.1236 


1.1302 


1.1369 


1.1436 


1 . 1503 


41 


49 


1.1503 


1.1571 


1 . 1639 


1.1708 


1.1777 


1.1847 


1.1917 


40 


50 


1.1917 


1.1988 


1.2059 


1.2131 


1.2203 


1.2275 


1 2349 


39 


51 


1.2349 


1.2422 


1.2496 


1.2571 


1.2647 


1.2723 


1.2799 


38 


52 


1.2799 


1.2876 


1.2954 


1.3032 


1.3111 


1.3190 


1.3270 


37 


53 


1.3270 


1.3351 


1.3432 


1.3514 


1.3596 


1.3680 


1.3763 


36 


54 


1.3763 


1.3848 


1.3933 


1.4019 


1.4106 


1.4193 


1.4281 


35 


55 


1.4281 


1.4370 


1.4459 


1.4550 


1.4641 


1.4733 


1.4825 


34 


56 


1.4825 


1.4919 


1.5013 


1.5108 


1.5204 


1.5301 


1.5398 


33 


57 


1.5398 


1.5497 


1.5596 


1.5696 


1.5798 


1.5900 


1.6003 


32 


58 


1.6003 


1.6107 


1.6212 


1.6318 


1.6435 


1.6533 


1.6642 


31 


59 


1.6642 


1.6753 


1.6864 


1.6976 


1.7090 


1.7204 


1.7320 


30 


60 


1.7320 


1.7437 


1.7555 


1 . 7674 


1.7795 


1.7917 


1.8040 


29 


61 


1.8040 


1.8164 


1.8290 


1.8417 


1.8546 


1.8676 


1.8807 


28 


63 


1.8807 


1.8940 


1.9074 


1.9209 


1.9347 


1 . 9485 


1.9626 


27 


63 


1.9626 


1.9768 


1.9911 


2.0056 


2.0303 


2.0352 


2.0503 


26 


64 


2.0503 


2.0655 


2.0809 


2.0965 


2.1123 


2.1283 


2.1445 


25 


65 


2.1445 


2.1609 


2.1774 


2.1943 


2.2113 


2.2285 


2.2400 


24 


66 


2.2460 


2.2637 


2.2816 


2.2998 


2.3183 


2.3369 


2.3558 


23 


67 


2.3558 


2.3750 


2.3944 


2.4142 


2.4342 


2.4545 


2.4750 


22 


68 


2.4750 


2.4959 


2.5171 


2.5386 


2.5604 


2.5826 


2.6050 


21 


69 


2.6350 


2.6379 


2.6510 


2.6746 


2.6985 


2.7228 


2.7474 


20 


70 


2.7474 


2.7725 


2.7980 


2.8239 


2.8503 


2.8770 


2.9042 


19 


71 


2.9042 


2.9318 


2.9600 


2.9886 


3.0178 


3.0474 


3.0776 


18 


72 


3.0776 


3.1084 


3.1397 


3.1715 


3.2040 


3.2371 


3.2708 


17 


73 


3.2708 


3.3052 


3.3402 


3.3759 


3.4133 


3.4495 


3.4874 


16 


74 


3.4874 


3.5260 


3.5655 


3.6058 


3.6470 


3.6890 


3.7330 


15 


75 


3.7330 


3.7759 


3.8208 


3.86G7 


3.9136 


3.9616 


4.0107 


14 


76 


4.0107 


4.0610 


4.1125 


4.1653 


4.2193 


4.2747 


4.3314 


13 


77 


4.3^14 


4.3896 


4.4494 


4.5107 


4.5736 


4.6382 


4.7046 


12 


78 


4.7046 


4.7728 


4.8430 


4.9151 


4.9894 


5 065S 


5.1445 


11 


79 


5.1445 


5.2256 


5.3092 


5.3955 


5.4845 


5.5763 


5.6712 


10 


80 


5.6712 


5.7693 


5.8708 


5.9757 


6.0844 


6.1970 


6.3137 


9 


81 


6.3137 


6.4348 


6.5605 


6.6911 


6.8369 


6.9682 


7.1153 


8 


82 


7.1153 


7.2687 


7.4287 


7.5957 


7.7703 


7.9530 


8.1443 


7 


83 


S.1443 


8.3449 


8.5555 


8.7768 


9.0098 


9.2553 


9.5143 


6 


81 


9.5143 


9.7881 


10.078 


10.385 


10.711 


11.059 


11.430 


5 


85 


11.430 


11.826 


12.250 


12.706 


13.196 


13.726 


14.300 


4 


88 


14.300 


14.924 


15.604 


16.349 


17.169 


18.075 


19.081 


3 


87 


19.081 


20.205 


21.470 


22.904 


24.541 


26.431 


28.636 


2 


88 


28.636 


31.241 


34.367 


38.188 


42.9o4 


49.103 


57.290 


1 


89 


57.290 


68.750 


85.939 


114.58 


171.88 


343.77 


00 


; 




60' 


50" 


40' 


30' 


20' 


10' 


C 


Deg. 



NATURAL COTANGENT. 



120 



BROWN & SHARPE MFG. CO. 



NATUKAL SECANT. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50 


60' 


i 





1.0000 


1.0000 


1.0000 


1.0000 


1.0000 


1.0001 


1.0001 


89 


1 


l.OOOL 


1.0002 


1.0002 


1.0008 


1.0001 


1.0005 


1.0006 


88 


2 


1.0006 


1.0007 


1 0008 


1.0009 


1.0010 


1.0012 


1.0013 


87 


3 


1.0013 


1.0015 


1.0016 


1.0018 


1.0020 


1.0022 


1.0024 


86 


4 


1.0024 


1.0026 


1.0028 


1.0080 


1.0033 


1.0035 


1.0038 


85 1 


5 


1.0088 


1.0040 


1.0043 


1 0016 


1.0049 


1.0052 


1.0055 


84 


6 


1.0055 


1.0058 


1.0081 


1.0061 


1.0068 


i.oori 


1.0075 


83 




1.0075 


1.0078 


1.0082 


1.0086 


1.0090 


1.0094 


1.0098 


82 


8 


1.0098 


1.0102 


1.0108 


1.0111 


1.0115 


1.0120 


1.0124 


81 


9 


1.0124 


1.0129 


1.0134 


1.0139 


1.0144 


1.0149 


1.0154 


80 


10 


1.0154 


1.0159 


1.0164 


1.0170 


1 . 0175 


1.0181 


1.0187 


79 


11 


1.0187 


1.0192 


1.0198 


1.0204 


1.0210 


1.0217 


J. 0223 


78 


12 


1.0223 


1.0229 


1.0286 


1.0242 


1.0249 


1.0256 


1.0263 


77 


13 


1.0263 


1.0269 


1.0277 


1.0284 


1.0291 


1.0298 


1.0308 


76 


14 


' 1.0303 


1.0313 


1.0821 


1.0329 


1.0336 


1.0344 


1.0352 


75 


15 


1.0352 


1.0360 


1.0369 


1.0877 


1.0385 


1.0394 


1.0402 


74 


IG 


1 1.0402 


1.0411 


1.0420 


1.0429 


1.0438 


1.0447 


1 . 0456 


73 


17 


1.0456 


1.0466 


1.0475 


1.0485 


1.0494 


1.0504 


1.0514 


72 


18 


1.0514 


1.0524 


1.0534 


1.0544 


1.0555 


1.0565 


1.0576 


71 


19 


1.0576 


1.0586 


1.0597 


1.0308 


1.0819 


1 0630 


1.0641 


70 


20 


1.0641 


1.0658 


1.0634 


1.0376 


1.0387 


1.0699 


1.0711 


69 


21 


1.0711 


1.0728 


1.0735 


1.0747 


1.0760 


1.0772 


1.0785 


68 


23 


1.0785 


1.0798 


1.0810 


1.0823 


1.0837 


1.0850 


1.0833 


67 


23 


1.0863 


1.0877 


1.0890 


1.0904 


1.0918 


1.0932 


1.0946 


m 


24 


1.0 J46 


1.0960 


1.0974 


1.0989 


1.1004 


1.1018 


1.1033 


65 


25 


1.1033 


1.1048 


1.1063 


1.1079 


1.1094 


1.1110 


1.1126 


64 


26 


1.1126 


1.1141 


1.1157 


1.1174 


1.1190 


1.1206 


1.1223 


63 


27 


1 1223 


1.1289 


1.1256 


1.1273 


1.1290 


1 . 1308 


1.1825 : 


62 


28 


1 . 1325 


1 . 1843 


1.1361 


1.1378 


1.1396 


1.1415 


1.1438 1 


61 


29 


1 . 1433 


1 . 1452 


1 . 1470 


1.1489 


1.1508 


1 . 1527 


1.1547 1 


60 


30 


1.1547 


1.1566 


1.1586 


1.1605 


1.1625 


1.1646 


1.1666 ' 


59 


3L 


1.1666 


1.1686 


1.1707 


1.172S 


1.1749 


1.1770 


1.1791 


58 


32 


1.1791 


1.1813 


1.1835 


1.1856 


1.1878 


1.1901 


1.1923 


57 


33 


1.1923 


1.1946 


1.19C9 


1.1992 


1 . 2015 


1.2038 


1.2062 


56 


34 


1.2032 


1.2085 


1.2109 


1.2134 


1.2158 


1.2182 


1.2207 


55 


35 


1.2207 


1.2282 


1.2257 


1.2283 


1.2308 


1.2334 


1.2360 


54 


33 


1.2860 


1.23S6 


1.2413 


1.2440 


1.2436 


1.2494 


1.2521 


53 


37 


1.2521 


1.254S 


1.2576 


1.2304 


1.2632 


1.2661 


1.2690 


52 


38 


1.2690 


1.2719 


1.2748 


1.2777 


1.2807 


1.2887 


1.2867 


51 


39 


1.2867 


1.2898 


1.2928 


1.2959 


1.2990 


1.3022 


1.3054 


50 


40 


1.3054 


1.3086 


1.3118 


1.3150 


1.3183 


1.8216 


1.3250 


49 


41 


1.3230 


1.3283 


1.3317 


1.3351 


1.3386 


1.342L 


1.3456 


48 


42 


1.3456 


1.3491 


1.3527 


1.3563 


1.3599 


1.3636 


1.3673 


47 


43 


1.3673 


1.3710 


1.3748 


1.3785 


1.3824 


1.3862 


1.3901 


46 


44 


1.3901 


1.3940 


1.3980 


1.4020 


1.4030 


1.4101 


1.4142 


45 




GO' 


50' 


40' 


30' 


20' 


10' 





Deg. 



NATtJKAL COSECANT. 



BROWN & SHARPE MFG. CO. 



121 



NATURAL SECANT. 



Deg. 


0- 


10' 


20' 


30' 


40' 


50' 


GO' 


1 


45 


1.4142 


1.4183 


1.4335 


1.4367 


1.4309 


1.4353 


1.4395 


44 


46 


1.4395 


1.4439 


1.4483 


1.4537 


1.4573 


1.4017 


1.4603 


1 43 


47 


1.4663 


1.4708 


1.4755 


1.4801 


1.4849 


1.4890 


1.4944 


42 


48 


1.4944 


1.4993 


1.5042 


1.5091 


1.5141 


1.5191 


1.5342 


41 


49 


1.5242 


1.5393 


1.5345 


1.5397 


1.5450 


1.5503 


1.5557 


40 


50 


1.5557 


1.5611 


1.5666 


1.5721 


1.5777 


1.5883 


1.5890 


39 


51 


1.5890 


1.5947 


1.6005 


1.0063 


1.6133 


1.6183 


1.6343 


38 


53 


1.6243 


1.6303 


1.6364 


1.6436 


1.6489 


1.6553 


1.0016 


37 


53 


1.6616 


1.6680 


1.6745 


1.0811 


1.0878 


1.0945 


1.7013 


36 


54 . 


1.7013 


1.7081 


1.7150 


1.7230 


1.7391 


1 . 7303 


1.7434 


35 


55 


1.7434 


1.7507 


1.7580 


1.7655 


1.7730 


1.7806 


1.78S3 


34 


56 


1.7883 


1.7900 


1.8038 


1.8118 


1.8198 


1.8378 


1.8360 


33 


57 


1.8360 


1.8443 


1.8527 


1.8611 


1.8097 


1.8783 


1.8870 


32 


58 


1.8870 


1.8959 


1.9048 


1.9138 


1.9330 


1.9323 


1.9416 


31 


59 


1.9416 


1.9510 


1.9606 


1.9703 


1.9800 


1 . 985)9 


2.0000 


30 


60 


3.0000 


3.0101 


2.0203 


3.0307 


3.0413 


3.0519 


2.0;i26 


29 


61 


3.063G 


3.0735 


2.0845 


3.0957 


3.1070 


2.1184 


2.1300 


28 


62 


3.1300 


3.1417 


2.1536 


3.1656 


3.1778 


2.1901 


2.2026 


27 


63 


3.3036 


3.3153 


2.2281 


3.3411 


3.3543 


2.2076 


2.2811 


26 


64 


3.3811 


3.3948 


3.3087 


3.3238 


3.3370 


2.3515 


2.3602 


25 


65 


2.3662 


3.3810 


3.3961 


3.4114 


3.4359 


2.4426 


2.4585 


24 


66 


2.4585 


3.4747 


3.4911 


3.5078 


3.5347 


3.5418 


2.5593 


23 


67 


2.55113 


3.57G9 


3.5949 


3.6131 


3.6316 


2.0503 


2.0094 


22 


68 


2.6694 


3.6883 


3.7085 


3.7385 


3.7488 


2.7094 


2.7904 


i 21 


69 


2.7904 


3.8117 


3.8834 


3.8554 


3.8778 


2.9006 


2.9238 


20 


70 


2.9338 


3.9473 


3.9713 


3.9957 


3.0205 


3.0458 


3.0715 


19 


71 


3.0715 


3.0977 


3.1343 


3.1515 


3.1791 


3.2073 


3.2360 


18 


73 


3.2360 


3.3653 


3.3951 


3.3355 


3.3564 


3.3880 


3.4203 


17 


73 


3.4303 


3.4531 


3.4867 


3.5309 


3.5558 


3.5915 


3.6279 


16 


74 


3.6379 


3.6651 


3.7031 


3.7419 


3.7816 


3.8332 


3.8637 


15 


75 


3.8(537 


3.9061 


3.9495 


3.9939 


4.0393 


4.0859 


4.1335 


14 


76 


4.1335 


4.1833 


4.3333 


4.3836 


4.3363 


4.3901 


4.4454 


13 


77 


4.4454 


4.5031 


4. 5004 


4.0303 


4.0816 


4.7448 


4.8097 


12 


78 


4.8097 


4.8764 


4.9451 


5.0158 


5.0886 


5.1035 


5.2408 


11 


79 


5.3408 


5.3304 


5.4026 


5.4874 


5.5749 


5 . 6653 


5.7587 


1 10 


80 


5.7587 


5.8553 


5.9553 


6.0588 


6.1600 


6.3771 


6.3924 


9 


81 


6.3924 


6.5130 


6.6363 


6.7654 


6.8997 


7.0396 


7.1852 


8 


83 


7.1853 


7. 3371 


7.4957 


7.6613 


7.8344 


8.0156 


8.2055 


7 


83 


8.3055 


8.4046 


8.6137 


8.8336 


9.0G51 


9.3091 


9.5667 


6 


84 


9.5667 


9.8391 


10.137 


10.433 


10.758 


11.104 


11.473 


5 


85 


11.473 


11.868 


13.3!)1 


13.745 


13.334 


13.763 


14.335 


4 


86 


14.335 


14.957 


15.036 


16.380 


17.198 


18.103 


19.107 


3 


87 


19.107 


30.230 


31.493 


33.935 


34.563 


36.4no 


28 653 


2 


88 


38.653 


31.357 


34.383 


38.201 


43 975 


49.114 


57.298 


1 


89 


57.398 


68.757 


85.945 


114.59 


171.88 


343.77 


00 







60' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COSECANT. 






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